WEBVTT 00:00:00.000 --> 00:00:02.444 align:middle line:90% [SQUEAKING] 00:00:02.444 --> 00:00:03.902 align:middle line:90% [RUSTLING] 00:00:03.902 --> 00:00:05.846 align:middle line:90% [CLICKING] 00:00:05.846 --> 00:00:15.566 align:middle line:90% 00:00:15.566 --> 00:00:18.040 align:middle line:84% JUEJUN HU: For this lecture, I'm going 00:00:18.040 --> 00:00:19.800 align:middle line:84% to try to do three different things. 00:00:19.800 --> 00:00:22.320 align:middle line:84% I will start by wrapping up about the essentially 00:00:22.320 --> 00:00:25.080 align:middle line:84% optical properties, optical applications. 00:00:25.080 --> 00:00:27.840 align:middle line:84% And then I will talk a little bit about the pset. 00:00:27.840 --> 00:00:29.733 align:middle line:84% And after that, I will actually discuss 00:00:29.733 --> 00:00:31.400 align:middle line:84% about different characterization methods 00:00:31.400 --> 00:00:33.662 align:middle line:84% that can be applied for amorphous materials. 00:00:33.662 --> 00:00:35.120 align:middle line:84% Specifically in this lecture, we're 00:00:35.120 --> 00:00:37.360 align:middle line:84% going to focus on diffraction techniques. 00:00:37.360 --> 00:00:39.060 align:middle line:84% All right, so in the last lecture, 00:00:39.060 --> 00:00:41.000 align:middle line:84% I showed you this video which essentially 00:00:41.000 --> 00:00:44.920 align:middle line:84% showed some developments of a higher refractive index glasses, 00:00:44.920 --> 00:00:47.540 align:middle line:84% specifically for augmented reality applications. 00:00:47.540 --> 00:00:51.120 align:middle line:84% So if you look at the key properties that 00:00:51.120 --> 00:00:54.520 align:middle line:84% are desirable for this kind of augmented reality glasses, 00:00:54.520 --> 00:00:57.640 align:middle line:84% as I said, high refractive index is basically the number 00:00:57.640 --> 00:00:58.660 align:middle line:90% one requirement there. 00:00:58.660 --> 00:01:02.000 align:middle line:84% So the reason high refractive index is needed, 00:01:02.000 --> 00:01:04.680 align:middle line:84% it basically goes back to essentially the field 00:01:04.680 --> 00:01:06.360 align:middle line:90% of view of the system. 00:01:06.360 --> 00:01:08.600 align:middle line:84% So we know that the refractive index essentially 00:01:08.600 --> 00:01:10.830 align:middle line:84% dictates the range of angles where the light can 00:01:10.830 --> 00:01:13.080 align:middle line:84% be trapped inside the waveguide through total internal 00:01:13.080 --> 00:01:13.660 align:middle line:90% reflection. 00:01:13.660 --> 00:01:19.120 align:middle line:84% So this is basically the angle theta. 00:01:19.120 --> 00:01:24.320 align:middle line:84% So basically, this angle theta has 00:01:24.320 --> 00:01:28.080 align:middle line:84% to be greater than the critical angle in order 00:01:28.080 --> 00:01:31.160 align:middle line:84% for the light to be able to trap via total internal reflection 00:01:31.160 --> 00:01:32.220 align:middle line:90% in the waveguide. 00:01:32.220 --> 00:01:36.120 align:middle line:84% So if the index is higher, this critical angle 00:01:36.120 --> 00:01:37.720 align:middle line:90% is going to get smaller. 00:01:37.720 --> 00:01:40.360 align:middle line:84% And that means that there's increased angular 00:01:40.360 --> 00:01:45.720 align:middle line:84% range where this waveguide can be used to trap light. 00:01:45.720 --> 00:01:49.960 align:middle line:84% So that actually translates in the end to a larger angular 00:01:49.960 --> 00:01:52.720 align:middle line:84% field of view where people can actually 00:01:52.720 --> 00:01:55.008 align:middle line:84% see images coming from the waveguide itself, 00:01:55.008 --> 00:01:56.800 align:middle line:84% because that's essentially a larger angular 00:01:56.800 --> 00:02:00.050 align:middle line:84% range of light rays that can propagate in the waveguide. 00:02:00.050 --> 00:02:02.267 align:middle line:84% And that's quite important for these AI applications, 00:02:02.267 --> 00:02:04.850 align:middle line:84% obviously, because we want to make sure that when you actually 00:02:04.850 --> 00:02:06.810 align:middle line:84% display something, you'll be able to feel 00:02:06.810 --> 00:02:09.130 align:middle line:84% like the large field of view, rather than only a very 00:02:09.130 --> 00:02:11.270 align:middle line:90% narrow, angular range. 00:02:11.270 --> 00:02:13.650 align:middle line:84% So that's why there has been a strong push 00:02:13.650 --> 00:02:18.450 align:middle line:84% towards high-index materials for this kind of AI applications. 00:02:18.450 --> 00:02:21.290 align:middle line:84% And you can see here that many of the glass vendors-- 00:02:21.290 --> 00:02:23.130 align:middle line:84% I'm using short as an example here. 00:02:23.130 --> 00:02:24.610 align:middle line:84% But many of the glass vendors also 00:02:24.610 --> 00:02:27.318 align:middle line:84% have similar glass products, now with a refractive index actually 00:02:27.318 --> 00:02:30.370 align:middle line:84% higher than 2.0, which is quite remarkable considering 00:02:30.370 --> 00:02:37.810 align:middle line:84% that most silicate oxide glasses only have index of about 1.5. 00:02:37.810 --> 00:02:39.430 align:middle line:84% All right, so that's one property. 00:02:39.430 --> 00:02:43.690 align:middle line:84% Now, other things, we also need high transmission 00:02:43.690 --> 00:02:44.750 align:middle line:90% for obvious reasons. 00:02:44.750 --> 00:02:47.730 align:middle line:84% You want to make sure there's minimal efficiency loss. 00:02:47.730 --> 00:02:50.930 align:middle line:84% Low birefringence-- birefringence 00:02:50.930 --> 00:02:53.610 align:middle line:84% refers to essentially anisotropy of its optical refractive index 00:02:53.610 --> 00:02:55.090 align:middle line:90% in the material. 00:02:55.090 --> 00:02:58.090 align:middle line:84% In a glass material, normally it should be isotropic. 00:02:58.090 --> 00:03:00.090 align:middle line:84% So there should be zero birefringent. 00:03:00.090 --> 00:03:02.130 align:middle line:84% But birefringence can be caused if there's 00:03:02.130 --> 00:03:04.530 align:middle line:84% any kind of residual stress in the material. 00:03:04.530 --> 00:03:06.410 align:middle line:84% So low birefringence, essentially, you 00:03:06.410 --> 00:03:08.970 align:middle line:84% need to process the material or annealing glass material 00:03:08.970 --> 00:03:11.853 align:middle line:84% sufficiently well to annihilate any kind of residual stress. 00:03:11.853 --> 00:03:13.770 align:middle line:84% And that will be important to prevent any kind 00:03:13.770 --> 00:03:17.210 align:middle line:90% of optical distortions. 00:03:17.210 --> 00:03:20.850 align:middle line:84% Homogeneity is to essentially lower 00:03:20.850 --> 00:03:24.010 align:middle line:84% any kind of image distortion, enhance image quality, 00:03:24.010 --> 00:03:25.810 align:middle line:90% and prevent scattering. 00:03:25.810 --> 00:03:28.470 align:middle line:84% Same thing-- flatness and thickness homogeneity, 00:03:28.470 --> 00:03:31.370 align:middle line:84% this is also helps to ensure that optical designers can 00:03:31.370 --> 00:03:33.730 align:middle line:84% know very well the parameters of the components, 00:03:33.730 --> 00:03:36.410 align:middle line:84% and therefore ensure that optical performance is 00:03:36.410 --> 00:03:38.930 align:middle line:90% consistent with all design. 00:03:38.930 --> 00:03:43.050 align:middle line:84% Roughness-- low roughness helps to reduce any kind of scattering 00:03:43.050 --> 00:03:45.930 align:middle line:84% loss, enhances transmission, and reduces any kind 00:03:45.930 --> 00:03:49.510 align:middle line:90% of haze coming from the glass. 00:03:49.510 --> 00:03:52.090 align:middle line:90% The last one here cosmetic-- 00:03:52.090 --> 00:03:56.250 align:middle line:84% you notice that the unit is a scratch / dig. 00:03:56.250 --> 00:03:58.690 align:middle line:84% So what this means-- the first number 00:03:58.690 --> 00:04:00.690 align:middle line:84% called scratch essentially gives you 00:04:00.690 --> 00:04:04.450 align:middle line:84% the maximum allowed width of a scratch on the finished glass 00:04:04.450 --> 00:04:06.290 align:middle line:90% surface in microns. 00:04:06.290 --> 00:04:10.210 align:middle line:84% So it means essentially that the maximum width 00:04:10.210 --> 00:04:14.130 align:middle line:84% of the scratch you can find on a glass surface is 40 microns. 00:04:14.130 --> 00:04:16.970 align:middle line:84% Dig, on the other hand, refers to essentially more 00:04:16.970 --> 00:04:20.450 align:middle line:84% like point defects, like pinholes, inclusions, 00:04:20.450 --> 00:04:22.910 align:middle line:90% maybe like surface. 00:04:22.910 --> 00:04:24.470 align:middle line:84% There's a little pit or something. 00:04:24.470 --> 00:04:28.850 align:middle line:90% So this number is in 10 microns. 00:04:28.850 --> 00:04:32.690 align:middle line:84% So 20 dig means that the maximum size of this point 00:04:32.690 --> 00:04:34.650 align:middle line:84% defect you find on the glass component surface 00:04:34.650 --> 00:04:36.173 align:middle line:90% is 200 microns maximum. 00:04:36.173 --> 00:04:37.090 align:middle line:90% That's what's allowed. 00:04:37.090 --> 00:04:40.530 align:middle line:84% So having a low scratch / dig number also 00:04:40.530 --> 00:04:42.510 align:middle line:84% ensures that you have high surface quality. 00:04:42.510 --> 00:04:44.330 align:middle line:84% And this minimizes any kind of loss 00:04:44.330 --> 00:04:47.312 align:middle line:84% due to essentially optical scattering at those defects. 00:04:47.312 --> 00:04:48.770 align:middle line:84% So those are the basic requirements 00:04:48.770 --> 00:04:51.130 align:middle line:84% for these kind of optical properties 00:04:51.130 --> 00:04:53.180 align:middle line:90% and morphology of the glass. 00:04:53.180 --> 00:04:57.500 align:middle line:84% For form factor, low wafer thickness 00:04:57.500 --> 00:05:01.700 align:middle line:84% combined with the high glass strength 00:05:01.700 --> 00:05:04.140 align:middle line:84% ensures that you can have a lightweight glass, which 00:05:04.140 --> 00:05:06.700 align:middle line:84% helps to reduce the weight for any kind of wearable devices. 00:05:06.700 --> 00:05:09.200 align:middle line:84% You also need to have enough mechanical strength 00:05:09.200 --> 00:05:11.740 align:middle line:84% so that these glass wafers can be handled 00:05:11.740 --> 00:05:13.620 align:middle line:84% in a high-throughput wafer processing 00:05:13.620 --> 00:05:16.940 align:middle line:84% systems, which is important for high-volume manufacturing. 00:05:16.940 --> 00:05:19.360 align:middle line:90% And finally, wafer size-- 00:05:19.360 --> 00:05:22.500 align:middle line:84% most of the state of the art fabrication facilities 00:05:22.500 --> 00:05:25.540 align:middle line:84% use wafer sizes of 200 or 300 millimeters, 8 or 12 inch 00:05:25.540 --> 00:05:29.660 align:middle line:84% wafers, which is compatible with what glass vendors can offer. 00:05:29.660 --> 00:05:32.500 align:middle line:84% So these are the key attributes of this kind of high index 00:05:32.500 --> 00:05:34.820 align:middle line:90% glasses for augmented reality. 00:05:34.820 --> 00:05:38.820 align:middle line:84% Now, if you look at specifically the chemistry 00:05:38.820 --> 00:05:41.380 align:middle line:84% that people use to realize high-index glasses, 00:05:41.380 --> 00:05:44.940 align:middle line:84% not surprisingly, the recipe here 00:05:44.940 --> 00:05:47.780 align:middle line:84% is to go with heavy metal oxides. 00:05:47.780 --> 00:05:49.460 align:middle line:84% As we have mentioned about lead, there 00:05:49.460 --> 00:05:52.900 align:middle line:84% are also alternatives-- non lead-containing oxides, 00:05:52.900 --> 00:05:55.300 align:middle line:84% such as zirconium oxide or titanium oxide. 00:05:55.300 --> 00:05:57.280 align:middle line:84% People can add them into the glass network 00:05:57.280 --> 00:05:58.780 align:middle line:84% to help increase the polarizability, 00:05:58.780 --> 00:06:00.300 align:middle line:90% and therefore refractive index. 00:06:00.300 --> 00:06:01.720 align:middle line:84% So if you look at this plot here, 00:06:01.720 --> 00:06:04.540 align:middle line:84% this is basically the [INAUDIBLE] plot 00:06:04.540 --> 00:06:06.200 align:middle line:84% for the different glass compositions. 00:06:06.200 --> 00:06:09.860 align:middle line:84% These glasses typically get crowded in this corner 00:06:09.860 --> 00:06:14.660 align:middle line:84% here, essentially with a large letter F. 00:06:14.660 --> 00:06:16.140 align:middle line:84% So these are filling glasses, which 00:06:16.140 --> 00:06:17.900 align:middle line:84% means that they are doped with heavy metal oxides 00:06:17.900 --> 00:06:19.280 align:middle line:90% to boost the refractive index. 00:06:19.280 --> 00:06:23.340 align:middle line:84% So that's the basic idea how you chemically formulate these glass 00:06:23.340 --> 00:06:26.260 align:middle line:90% materials. 00:06:26.260 --> 00:06:30.123 align:middle line:84% All right, so just to quickly summarize, 00:06:30.123 --> 00:06:32.540 align:middle line:84% we have talked about essentially optical fibers and planar 00:06:32.540 --> 00:06:33.777 align:middle line:90% waveguides here. 00:06:33.777 --> 00:06:35.860 align:middle line:84% For fibers, you need to understand the fabrication 00:06:35.860 --> 00:06:37.060 align:middle line:90% process. 00:06:37.060 --> 00:06:42.580 align:middle line:84% And for waveguides, it's important to understand the loss 00:06:42.580 --> 00:06:45.180 align:middle line:84% mechanisms in this kind of waveguide devices, 00:06:45.180 --> 00:06:48.780 align:middle line:84% and what are some of the key applications for waveguides 00:06:48.780 --> 00:06:50.900 align:middle line:90% in fiber devices. 00:06:50.900 --> 00:06:58.180 align:middle line:84% All right, so now what I will do is switch gears a little bit. 00:06:58.180 --> 00:07:04.820 align:middle line:84% And let's take a look at the pset. 00:07:04.820 --> 00:07:08.780 align:middle line:90% 00:07:08.780 --> 00:07:13.220 align:middle line:84% All right, so the first question is basically, 00:07:13.220 --> 00:07:15.700 align:middle line:84% just essentially look at this figure 00:07:15.700 --> 00:07:21.900 align:middle line:84% here and try to calculate the strain, as well 00:07:21.900 --> 00:07:24.940 align:middle line:84% as the temperature, at which the thermal stress starts 00:07:24.940 --> 00:07:25.940 align:middle line:90% to build up. 00:07:25.940 --> 00:07:28.250 align:middle line:84% So what I mean here is essentially, 00:07:28.250 --> 00:07:29.500 align:middle line:90% you just look at this picture. 00:07:29.500 --> 00:07:31.042 align:middle line:84% From the picture, you can essentially 00:07:31.042 --> 00:07:33.780 align:middle line:84% read out the parameters like the bending radius, 00:07:33.780 --> 00:07:35.900 align:middle line:84% as well as the thickness of the wire, 00:07:35.900 --> 00:07:37.940 align:middle line:84% using some kind of image processing software. 00:07:37.940 --> 00:07:41.140 align:middle line:84% And then you can plug into this basically this bimetallic strip 00:07:41.140 --> 00:07:42.740 align:middle line:90% equation. 00:07:42.740 --> 00:07:46.850 align:middle line:84% From the equation, if the radius of this band here, 00:07:46.850 --> 00:07:49.710 align:middle line:84% you can then use that to actually calculate the strain 00:07:49.710 --> 00:07:51.310 align:middle line:90% mismatch, as well as delta T. 00:07:51.310 --> 00:07:54.450 align:middle line:84% So I'm not going to go through the details here. 00:07:54.450 --> 00:07:56.590 align:middle line:84% I think you all get the basic steps. 00:07:56.590 --> 00:07:59.350 align:middle line:84% The numbers may not be the same as what I'm showing here, 00:07:59.350 --> 00:08:02.030 align:middle line:84% because it turns out everyone has slightly different readings 00:08:02.030 --> 00:08:04.190 align:middle line:84% in terms of how thick the wire is, 00:08:04.190 --> 00:08:05.530 align:middle line:90% as well as the bending radius. 00:08:05.530 --> 00:08:07.810 align:middle line:84% So there's a pretty large error bar associated with that. 00:08:07.810 --> 00:08:09.790 align:middle line:84% So that's why I don't worry too much about the specific numbers 00:08:09.790 --> 00:08:10.390 align:middle line:90% there. 00:08:10.390 --> 00:08:12.250 align:middle line:84% As long as you get the method right, 00:08:12.250 --> 00:08:14.230 align:middle line:90% I give you full credit for that. 00:08:14.230 --> 00:08:16.710 align:middle line:90% So I did my fitting. 00:08:16.710 --> 00:08:18.590 align:middle line:84% This is probably not very accurate. 00:08:18.590 --> 00:08:22.430 align:middle line:84% I roughly guess that the thickness is about 1 millimeter. 00:08:22.430 --> 00:08:26.890 align:middle line:84% And this actually gives a delta T of 926 degrees Celsius, 00:08:26.890 --> 00:08:29.990 align:middle line:84% which means that the strains start 00:08:29.990 --> 00:08:31.990 align:middle line:90% to build up at 950 Celsius. 00:08:31.990 --> 00:08:34.201 align:middle line:84% It falls somewhere within the working range. 00:08:34.201 --> 00:08:35.909 align:middle line:84% So it's somewhat more or less reasonable. 00:08:35.909 --> 00:08:38.789 align:middle line:84% You expect the strain start to build up when the glass loses 00:08:38.789 --> 00:08:40.710 align:middle line:90% the ability to deform. 00:08:40.710 --> 00:08:44.190 align:middle line:84% So you expect that temperature to be falling in maybe somewhere 00:08:44.190 --> 00:08:46.328 align:middle line:84% around the working range, and maybe at the lower 00:08:46.328 --> 00:08:47.370 align:middle line:90% end of the working range. 00:08:47.370 --> 00:08:49.630 align:middle line:84% But I do want to mention, there's 00:08:49.630 --> 00:08:51.990 align:middle line:84% a pretty large error bar associated with this number 00:08:51.990 --> 00:08:52.990 align:middle line:90% here. 00:08:52.990 --> 00:08:54.370 align:middle line:90% This comes from two parts here. 00:08:54.370 --> 00:08:56.850 align:middle line:84% Possibly one is actually the thickness variation. 00:08:56.850 --> 00:08:59.110 align:middle line:84% Two is that we also made the assumption 00:08:59.110 --> 00:09:01.270 align:middle line:84% that the borosilicate and the soda-lime 00:09:01.270 --> 00:09:02.635 align:middle line:90% has exactly the same width. 00:09:02.635 --> 00:09:04.510 align:middle line:84% From the experiment, you know that's actually 00:09:04.510 --> 00:09:05.730 align:middle line:90% not exactly the case. 00:09:05.730 --> 00:09:08.190 align:middle line:84% One of them is definitely wider than the other. 00:09:08.190 --> 00:09:10.270 align:middle line:84% So I also did some more simulations. 00:09:10.270 --> 00:09:15.150 align:middle line:84% Actually, this is basically a Matlab code I used for that. 00:09:15.150 --> 00:09:16.410 align:middle line:90% There are two plots here. 00:09:16.410 --> 00:09:20.190 align:middle line:84% So the first one, this is if we assume the two pieces of glass 00:09:20.190 --> 00:09:23.830 align:middle line:84% have the same thickness, but you have essentially different 00:09:23.830 --> 00:09:25.175 align:middle line:90% thickness values. 00:09:25.175 --> 00:09:26.550 align:middle line:84% You see that the delta T actually 00:09:26.550 --> 00:09:28.550 align:middle line:84% linearly scales with thickness, which makes sense. 00:09:28.550 --> 00:09:30.030 align:middle line:84% The thicker it is, the more strain 00:09:30.030 --> 00:09:33.030 align:middle line:90% you need to actually bend it. 00:09:33.030 --> 00:09:36.148 align:middle line:84% Also, some of the minor impacts come into effect 00:09:36.148 --> 00:09:37.690 align:middle line:84% if you have varying thickness ratios. 00:09:37.690 --> 00:09:39.550 align:middle line:84% In this case, I keep the total thickness 00:09:39.550 --> 00:09:43.000 align:middle line:84% to be 2 millimeters by varying the thickness ratios. 00:09:43.000 --> 00:09:46.140 align:middle line:84% You see that the thickness ratio again, when it's close to unity, 00:09:46.140 --> 00:09:48.440 align:middle line:84% you get actually more or less the minimum delta T. 00:09:48.440 --> 00:09:50.300 align:middle line:84% But if you have a different thickness ratio, 00:09:50.300 --> 00:09:52.300 align:middle line:84% you can imagine that it gets more difficult to bend it. 00:09:52.300 --> 00:09:54.780 align:middle line:84% If you have a thinner piece, try to bend the thicker piece. 00:09:54.780 --> 00:09:56.798 align:middle line:84% So that's why the temperature also increases. 00:09:56.798 --> 00:09:58.340 align:middle line:84% But it has a relatively minor impact. 00:09:58.340 --> 00:10:01.320 align:middle line:84% So the main error actually comes from the fact that we cannot 00:10:01.320 --> 00:10:03.800 align:middle line:90% read the thickness precisely. 00:10:03.800 --> 00:10:07.202 align:middle line:84% So that's something we have to live with basically. 00:10:07.202 --> 00:10:08.660 align:middle line:84% All right, so that's the first one. 00:10:08.660 --> 00:10:13.600 align:middle line:84% The second one, this one is asking about 00:10:13.600 --> 00:10:15.880 align:middle line:84% whether we can come up with-- or essentially, 00:10:15.880 --> 00:10:20.212 align:middle line:84% this one is about the microscopic entities 00:10:20.212 --> 00:10:22.420 align:middle line:84% contributing to dielectric constant refractive index. 00:10:22.420 --> 00:10:23.780 align:middle line:90% That's quite straightforward. 00:10:23.780 --> 00:10:27.880 align:middle line:84% Essentially, depending on the frequency, 00:10:27.880 --> 00:10:30.160 align:middle line:84% if it's a relatively low frequency, 00:10:30.160 --> 00:10:34.360 align:middle line:84% like microwave, then all the atoms, all the ions, 00:10:34.360 --> 00:10:37.560 align:middle line:84% contribute to polarizability, as well as their electrons. 00:10:37.560 --> 00:10:39.182 align:middle line:84% But if you go to high frequencies, 00:10:39.182 --> 00:10:40.640 align:middle line:84% like optical frequencies, then it's 00:10:40.640 --> 00:10:42.473 align:middle line:84% only the electrons that can actually respond 00:10:42.473 --> 00:10:44.600 align:middle line:90% to oscillating electric fields. 00:10:44.600 --> 00:10:46.920 align:middle line:84% And then for this equation, this equation 00:10:46.920 --> 00:10:49.580 align:middle line:84% actually applies for all the non-magnetic materials. 00:10:49.580 --> 00:10:52.720 align:middle line:84% It's quite generic provided that you measure the epsilon 00:10:52.720 --> 00:10:54.880 align:middle line:84% r as well as n at the same frequency 00:10:54.880 --> 00:10:58.080 align:middle line:90% or the same wavelength. 00:10:58.080 --> 00:11:01.280 align:middle line:84% All right, so now for the next question here, 00:11:01.280 --> 00:11:03.400 align:middle line:84% this one is about trying to come up 00:11:03.400 --> 00:11:06.160 align:middle line:84% with a glass material that potentially has 00:11:06.160 --> 00:11:09.540 align:middle line:90% an even lower loss than silica. 00:11:09.540 --> 00:11:13.160 align:middle line:90% 00:11:13.160 --> 00:11:16.980 align:middle line:84% Some of you cited ZBLAN, the fluoride glass. 00:11:16.980 --> 00:11:17.640 align:middle line:90% I forgot. 00:11:17.640 --> 00:11:18.487 align:middle line:90% One of you, I think. 00:11:18.487 --> 00:11:19.320 align:middle line:90% [INTERPOSING VOICES] 00:11:19.320 --> 00:11:20.880 align:middle line:90% JUEJUN HU: Someone. 00:11:20.880 --> 00:11:22.680 align:middle line:90% OK. 00:11:22.680 --> 00:11:25.860 align:middle line:84% I mean, if you look at some of the online resources or papers, 00:11:25.860 --> 00:11:29.080 align:middle line:84% they claim that ZBLAN can have a very low minimum source. 00:11:29.080 --> 00:11:31.640 align:middle line:90% I think it's unvalidated. 00:11:31.640 --> 00:11:34.840 align:middle line:84% First of all, nobody really achieved it experimentally. 00:11:34.840 --> 00:11:38.218 align:middle line:84% Second of all, I always question whether or not 00:11:38.218 --> 00:11:40.760 align:middle line:84% you can get to that low level of loss given that there are so 00:11:40.760 --> 00:11:41.880 align:middle line:90% many different ion species. 00:11:41.880 --> 00:11:43.672 align:middle line:84% And there's definitely contributions coming 00:11:43.672 --> 00:11:45.120 align:middle line:90% from concentration scattering. 00:11:45.120 --> 00:11:49.765 align:middle line:84% So my preference here would be let's look 00:11:49.765 --> 00:11:51.140 align:middle line:84% at the different sources of loss. 00:11:51.140 --> 00:11:53.907 align:middle line:84% If you look at what determines the minimum loss window, 00:11:53.907 --> 00:11:56.240 align:middle line:84% it pretty much comes from Rayleigh scattering and phonon 00:11:56.240 --> 00:11:56.740 align:middle line:90% absorption. 00:11:56.740 --> 00:11:59.040 align:middle line:84% These are the two mechanisms coming 00:11:59.040 --> 00:12:01.520 align:middle line:84% from high and low wavelengths that bound the minimum loss 00:12:01.520 --> 00:12:02.060 align:middle line:90% window. 00:12:02.060 --> 00:12:05.660 align:middle line:84% So if you want to reduce the loss, what you have to do, 00:12:05.660 --> 00:12:08.520 align:middle line:84% number one, you need to have low Rayleigh scattering loss. 00:12:08.520 --> 00:12:12.182 align:middle line:84% So for low Rayleigh scattering loss, the first thing, 00:12:12.182 --> 00:12:13.640 align:middle line:84% you want to have a single component 00:12:13.640 --> 00:12:15.640 align:middle line:84% glass, which allows you to eliminate 00:12:15.640 --> 00:12:20.058 align:middle line:84% any of the concentration scattering. 00:12:20.058 --> 00:12:21.600 align:middle line:84% There are some other techniques which 00:12:21.600 --> 00:12:24.520 align:middle line:84% I'm going to talk about in a second. 00:12:24.520 --> 00:12:26.660 align:middle line:84% Well, maybe I'll just mention it here. 00:12:26.660 --> 00:12:29.320 align:middle line:90% So that's one. 00:12:29.320 --> 00:12:32.000 align:middle line:84% The second one is that you also want 00:12:32.000 --> 00:12:35.730 align:middle line:84% to decrease the effective temperature of the glass 00:12:35.730 --> 00:12:36.792 align:middle line:90% materials. 00:12:36.792 --> 00:12:39.250 align:middle line:84% If you remember, when I was talking about the equation that 00:12:39.250 --> 00:12:41.070 align:middle line:84% actually gives you the Rayleigh scattering loss, 00:12:41.070 --> 00:12:43.610 align:middle line:84% I said that loss actually scales with fictive temperature. 00:12:43.610 --> 00:12:47.170 align:middle line:84% Fictive temperature is a measure of structure inhomogeneity. 00:12:47.170 --> 00:12:49.088 align:middle line:84% It's essentially a structural parameter. 00:12:49.088 --> 00:12:51.130 align:middle line:84% So the higher the fictive temperature, the higher 00:12:51.130 --> 00:12:53.505 align:middle line:84% the structural fluctuation, and therefore higher Rayleigh 00:12:53.505 --> 00:12:54.088 align:middle line:90% scattering. 00:12:54.088 --> 00:12:56.130 align:middle line:84% So if you're able to control the glass structure, 00:12:56.130 --> 00:12:58.710 align:middle line:84% make it essentially have a low fictive temperature, 00:12:58.710 --> 00:13:01.390 align:middle line:84% make it essentially closer to the equilibrium state, 00:13:01.390 --> 00:13:05.783 align:middle line:84% then you actually have a lower scattering loss. 00:13:05.783 --> 00:13:07.450 align:middle line:84% I want to mention that there is actually 00:13:07.450 --> 00:13:09.010 align:middle line:90% a very interesting paper. 00:13:09.010 --> 00:13:10.850 align:middle line:84% What I'm showing you here, this one's 00:13:10.850 --> 00:13:14.410 align:middle line:90% from last year's OFC conference. 00:13:14.410 --> 00:13:18.690 align:middle line:84% A group in Japan basically reported a record low silica 00:13:18.690 --> 00:13:24.875 align:middle line:84% fiber propagation loss with the loss down to a 0.14dB/km. 00:13:24.875 --> 00:13:27.250 align:middle line:84% This is basically a plot that shows there are two curves, 00:13:27.250 --> 00:13:30.070 align:middle line:84% shows for [? R&D ?] lab-grade optical fibers, 00:13:30.070 --> 00:13:31.630 align:middle line:90% how the loss evolves over time. 00:13:31.630 --> 00:13:36.530 align:middle line:84% And this is the lowest loss in industry scale optical fibers, 00:13:36.530 --> 00:13:39.190 align:middle line:84% how it decreases over time for silica-based optical fibers. 00:13:39.190 --> 00:13:42.670 align:middle line:84% Now, the trick they do there is that one 00:13:42.670 --> 00:13:45.090 align:middle line:90% is called the fiber core glass. 00:13:45.090 --> 00:13:47.703 align:middle line:84% We mentioned that to create index contrast for fiber. 00:13:47.703 --> 00:13:48.870 align:middle line:90% There are two ways to do it. 00:13:48.870 --> 00:13:50.828 align:middle line:84% You can either dope the core to increase index, 00:13:50.828 --> 00:13:52.650 align:middle line:84% or you dope the cladding to decrease index. 00:13:52.650 --> 00:13:55.370 align:middle line:84% Which one do you think is preferred if you 00:13:55.370 --> 00:13:56.910 align:middle line:90% want to really lower loss? 00:13:56.910 --> 00:13:59.370 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:13:59.370 --> 00:14:02.850 align:middle line:84% JUEJUN HU: Exactly, yeah, because the optical mode 00:14:02.850 --> 00:14:05.530 align:middle line:84% or the optical field mostly is confined to the core, 00:14:05.530 --> 00:14:06.550 align:middle line:90% less in the cladding. 00:14:06.550 --> 00:14:07.970 align:middle line:84% If you're doping the core, essentially, that 00:14:07.970 --> 00:14:09.890 align:middle line:84% means you're creating this kind of [? coherent ?] scattering 00:14:09.890 --> 00:14:10.310 align:middle line:90% there. 00:14:10.310 --> 00:14:10.970 align:middle line:90% So that's why. 00:14:10.970 --> 00:14:12.150 align:middle line:90% That's exactly what they do. 00:14:12.150 --> 00:14:13.690 align:middle line:90% The core is almost pure silica. 00:14:13.690 --> 00:14:15.650 align:middle line:84% They dope the cladding with fluorine, 00:14:15.650 --> 00:14:19.090 align:middle line:84% which is, of course, a very light element with very 00:14:19.090 --> 00:14:20.590 align:middle line:90% strong electronegativity. 00:14:20.590 --> 00:14:23.437 align:middle line:84% So they tend to have a very low electron polarizability. 00:14:23.437 --> 00:14:25.270 align:middle line:84% So only by putting a little bit of fluorine, 00:14:25.270 --> 00:14:27.270 align:middle line:84% they were able to create enough in the contrast, 00:14:27.270 --> 00:14:29.530 align:middle line:84% and therefore minimize [? coherent ?] scattering. 00:14:29.530 --> 00:14:31.223 align:middle line:90% They also played some tricks. 00:14:31.223 --> 00:14:33.390 align:middle line:84% Essentially, what they concluded is that in the end, 00:14:33.390 --> 00:14:34.932 align:middle line:84% they verified that their glasses also 00:14:34.932 --> 00:14:37.523 align:middle line:84% have exceptionally low fictive temperature. 00:14:37.523 --> 00:14:39.190 align:middle line:84% So they have low structural homogeneity. 00:14:39.190 --> 00:14:40.648 align:middle line:84% That's why they get to that number. 00:14:40.648 --> 00:14:41.870 align:middle line:90% Anyway, it comes back here. 00:14:41.870 --> 00:14:43.850 align:middle line:84% If you're looking for alternative compositions, 00:14:43.850 --> 00:14:46.650 align:middle line:84% of course, as I said, you want a single-component glass. 00:14:46.650 --> 00:14:49.650 align:middle line:84% You also want to try to lower the phonon absorption. 00:14:49.650 --> 00:14:52.655 align:middle line:84% So an effective way to do it is to go to heavy element glasses, 00:14:52.655 --> 00:14:54.530 align:middle line:84% which allow you to push this line essentially 00:14:54.530 --> 00:14:56.405 align:middle line:84% to the right-hand side, and therefore push it 00:14:56.405 --> 00:14:57.710 align:middle line:90% to a lower minimum loss window. 00:14:57.710 --> 00:15:00.010 align:middle line:84% So that's why one potential component 00:15:00.010 --> 00:15:02.650 align:middle line:90% could be a germanium oxide. 00:15:02.650 --> 00:15:04.750 align:middle line:84% Or you can go to things like germanium sulfide. 00:15:04.750 --> 00:15:07.018 align:middle line:84% Although I have already mentioned for chalcogenides, 00:15:07.018 --> 00:15:08.810 align:middle line:84% they may have different structural moieties 00:15:08.810 --> 00:15:09.638 align:middle line:90% within the network. 00:15:09.638 --> 00:15:10.430 align:middle line:90% So it may not work. 00:15:10.430 --> 00:15:13.730 align:middle line:84% So germanium oxide probably is the best option specifically 00:15:13.730 --> 00:15:15.010 align:middle line:90% for this problem here. 00:15:15.010 --> 00:15:18.070 align:middle line:84% But yeah, as long as you get the main idea right, 00:15:18.070 --> 00:15:22.370 align:middle line:84% that's what I'm really looking for here. 00:15:22.370 --> 00:15:27.370 align:middle line:84% All right, last one, comparing polysilicon and amorphous 00:15:27.370 --> 00:15:28.250 align:middle line:90% silicon. 00:15:28.250 --> 00:15:31.300 align:middle line:84% For any kind of electronic properties, whether solar cells 00:15:31.300 --> 00:15:35.940 align:middle line:84% or transistors, you want to have high mobility, long carrier 00:15:35.940 --> 00:15:39.740 align:middle line:84% lifetime in those cases, and less degradation. 00:15:39.740 --> 00:15:41.660 align:middle line:84% So in those cases, polycrystalline silicon 00:15:41.660 --> 00:15:42.740 align:middle line:90% is usually better. 00:15:42.740 --> 00:15:44.140 align:middle line:84% For waveguides, amorphous silicon 00:15:44.140 --> 00:15:46.540 align:middle line:84% is better because you don't have all the grains. 00:15:46.540 --> 00:15:49.660 align:middle line:84% And crystalline grains can contribute very significantly 00:15:49.660 --> 00:15:50.980 align:middle line:90% to scattering loss. 00:15:50.980 --> 00:15:54.500 align:middle line:84% In fact, in practice, people make polycrystalline silicon 00:15:54.500 --> 00:15:58.620 align:middle line:84% waveguides with a loss of the order like 10db/cm versus less 00:15:58.620 --> 00:16:01.320 align:middle line:84% than 1db/cm when it comes to amorphous silicon waveguides, 00:16:01.320 --> 00:16:03.640 align:middle line:84% because there is no crystalline grain structures. 00:16:03.640 --> 00:16:06.220 align:middle line:84% So potentially, it can be much lower loss. 00:16:06.220 --> 00:16:12.820 align:middle line:84% OK, any more questions about the pset here before we move on? 00:16:12.820 --> 00:16:14.580 align:middle line:90% All good, OK. 00:16:14.580 --> 00:16:19.700 align:middle line:84% All right, now let's go to the last section 00:16:19.700 --> 00:16:26.280 align:middle line:84% before we talk about the topic you proposed, 00:16:26.280 --> 00:16:28.220 align:middle line:84% which is biomineralization of amorphous 00:16:28.220 --> 00:16:30.620 align:middle line:84% materials, something I'm still making here. 00:16:30.620 --> 00:16:32.840 align:middle line:84% So in this section, I'm going to essentially talk 00:16:32.840 --> 00:16:35.340 align:middle line:84% about a few different techniques that people use to actually 00:16:35.340 --> 00:16:37.415 align:middle line:90% characterize amorphous state. 00:16:37.415 --> 00:16:39.040 align:middle line:84% Before I go into more specific details, 00:16:39.040 --> 00:16:42.540 align:middle line:84% I do have a few tables here that summarize 00:16:42.540 --> 00:16:44.620 align:middle line:84% some of the common methods you use 00:16:44.620 --> 00:16:48.220 align:middle line:84% to characterize different attributes of amorphous or glass 00:16:48.220 --> 00:16:48.940 align:middle line:90% material. 00:16:48.940 --> 00:16:51.240 align:middle line:90% So I won't go through it. 00:16:51.240 --> 00:16:55.020 align:middle line:84% You can just read through the table yourself. 00:16:55.020 --> 00:16:58.643 align:middle line:84% But what I just want to mention here is that if you look at-- 00:16:58.643 --> 00:17:00.060 align:middle line:84% so for example, one specific thing 00:17:00.060 --> 00:17:04.980 align:middle line:84% I want to quickly mention here if you go to optical properties. 00:17:04.980 --> 00:17:07.780 align:middle line:84% So for optical Properties, one of the very important metrics 00:17:07.780 --> 00:17:10.280 align:middle line:84% essentially is the loss or attenuation in the material. 00:17:10.280 --> 00:17:13.220 align:middle line:84% If you look at the techniques I'm listing here, a lot of them 00:17:13.220 --> 00:17:15.180 align:middle line:84% essentially mention that you can actually use 00:17:15.180 --> 00:17:17.060 align:middle line:90% measured optical attenuation. 00:17:17.060 --> 00:17:19.460 align:middle line:84% However, there's one very important difference here. 00:17:19.460 --> 00:17:21.579 align:middle line:84% So there are basically two kinds of methods 00:17:21.579 --> 00:17:23.859 align:middle line:90% to measure optical attenuation. 00:17:23.859 --> 00:17:29.700 align:middle line:84% One type is like ellipsometry, or maybe UV-Vis spectroscopy. 00:17:29.700 --> 00:17:33.380 align:middle line:84% So if you have a bulk material, that's 00:17:33.380 --> 00:17:34.680 align:middle line:90% actually quite straightforward. 00:17:34.680 --> 00:17:36.780 align:middle line:84% But if you have a thin field, which 00:17:36.780 --> 00:17:39.340 align:middle line:84% is what people use, for example, for waveguides, then 00:17:39.340 --> 00:17:40.940 align:middle line:84% when you're trying to use ellipsometry 00:17:40.940 --> 00:17:45.060 align:middle line:84% or UV-Vis spectroscopy to measure the absorption, 00:17:45.060 --> 00:17:47.540 align:middle line:84% you essentially perform more or less this kind 00:17:47.540 --> 00:17:50.100 align:middle line:90% of surface normal measurement. 00:17:50.100 --> 00:17:52.287 align:middle line:84% You essentially shine light from free space. 00:17:52.287 --> 00:17:54.620 align:middle line:84% You go through a thin film, you measure the transmission 00:17:54.620 --> 00:17:56.080 align:middle line:90% or you measure a reflection. 00:17:56.080 --> 00:17:58.660 align:middle line:84% And you use this information to actually fit 00:17:58.660 --> 00:18:00.580 align:middle line:90% the attenuation in the field. 00:18:00.580 --> 00:18:05.020 align:middle line:84% Now, if you look at the path length 00:18:05.020 --> 00:18:08.037 align:middle line:84% of light in this kind of measurement here, 00:18:08.037 --> 00:18:09.120 align:middle line:90% it's actually quite small. 00:18:09.120 --> 00:18:11.733 align:middle line:84% It's basically related to the thickness of the film, which 00:18:11.733 --> 00:18:13.900 align:middle line:84% is of order of maybe a few hundred nanometers, maybe 00:18:13.900 --> 00:18:15.340 align:middle line:90% a few microns. 00:18:15.340 --> 00:18:17.880 align:middle line:84% So this is one class of technique. 00:18:17.880 --> 00:18:21.740 align:middle line:84% This low short path length also means that intrinsically, 00:18:21.740 --> 00:18:26.510 align:middle line:84% it cannot measure films with very low absorption 00:18:26.510 --> 00:18:30.183 align:middle line:84% because the attenuation induced by propagation within such 00:18:30.183 --> 00:18:32.350 align:middle line:84% a short distance becomes negligible if it's very low 00:18:32.350 --> 00:18:32.850 align:middle line:90% loss. 00:18:32.850 --> 00:18:35.430 align:middle line:90% 00:18:35.430 --> 00:18:38.230 align:middle line:84% Versus there's another class of techniques 00:18:38.230 --> 00:18:41.710 align:middle line:84% that rely on in-plane propagation of light. 00:18:41.710 --> 00:18:45.130 align:middle line:90% So in this case, it can be a-- 00:18:45.130 --> 00:18:48.047 align:middle line:84% I listed prism coupling or I listed waveguide measurement. 00:18:48.047 --> 00:18:49.630 align:middle line:84% But basically, in most cases, the idea 00:18:49.630 --> 00:18:52.990 align:middle line:84% is that you want to couple light into the thin film 00:18:52.990 --> 00:18:55.630 align:middle line:84% and let it propagate inside the thin film in the form 00:18:55.630 --> 00:18:57.790 align:middle line:90% of essentially waveguide mode. 00:18:57.790 --> 00:19:01.630 align:middle line:84% So in this case, the path length can actually be a lot longer. 00:19:01.630 --> 00:19:05.590 align:middle line:84% It can be, like, millimeters or centimeters. 00:19:05.590 --> 00:19:07.650 align:middle line:84% So because of the much longer path length, 00:19:07.650 --> 00:19:11.670 align:middle line:84% you are able to actually measure much smaller optical absorption 00:19:11.670 --> 00:19:14.750 align:middle line:90% or scattering losses. 00:19:14.750 --> 00:19:17.230 align:middle line:84% So that's very important information 00:19:17.230 --> 00:19:20.662 align:middle line:84% I want to mention here, because it's quite common that you see 00:19:20.662 --> 00:19:22.870 align:middle line:84% people make the mistake, saying that they essentially 00:19:22.870 --> 00:19:24.590 align:middle line:84% did the absorption measurement, seeing 00:19:24.590 --> 00:19:27.350 align:middle line:84% that the K, or the imaginary part of the refractive index, 00:19:27.350 --> 00:19:28.130 align:middle line:90% is small enough. 00:19:28.130 --> 00:19:30.547 align:middle line:84% And they claim that this is actually transparent material. 00:19:30.547 --> 00:19:31.730 align:middle line:90% That's not always the case. 00:19:31.730 --> 00:19:33.438 align:middle line:84% If you want to verify that a material can 00:19:33.438 --> 00:19:35.950 align:middle line:84% be used for fiber or for waveguides, 00:19:35.950 --> 00:19:38.910 align:middle line:84% practically you have to make a fiber wave out of it 00:19:38.910 --> 00:19:39.950 align:middle line:90% and verify it. 00:19:39.950 --> 00:19:42.790 align:middle line:84% It's very difficult to verify the low-loss characteristics out 00:19:42.790 --> 00:19:45.870 align:middle line:90% of any other techniques. 00:19:45.870 --> 00:19:48.990 align:middle line:84% All right, so this is one thing I want to mention quickly here. 00:19:48.990 --> 00:19:53.230 align:middle line:84% Now next let's start by talking about diffraction 00:19:53.230 --> 00:19:56.050 align:middle line:84% here, both X-ray electron and neutron diffraction. 00:19:56.050 --> 00:20:00.230 align:middle line:84% So different techniques is something that people very 00:20:00.230 --> 00:20:02.630 align:middle line:84% routinely use to characterize material structures, 00:20:02.630 --> 00:20:05.088 align:middle line:84% which I believe you probably have also learned a little bit 00:20:05.088 --> 00:20:09.710 align:middle line:84% in either 3.091 or maybe a 3.010, I think. 00:20:09.710 --> 00:20:10.870 align:middle line:90% Yeah, OK, good. 00:20:10.870 --> 00:20:13.330 align:middle line:84% So we'll be essentially expanding a little bit on that, 00:20:13.330 --> 00:20:15.750 align:middle line:84% but focus specifically on characterization 00:20:15.750 --> 00:20:16.970 align:middle line:90% of amorphous materials. 00:20:16.970 --> 00:20:20.590 align:middle line:84% But just to start with, I'm showing you this famous picture 00:20:20.590 --> 00:20:21.750 align:middle line:90% here. 00:20:21.750 --> 00:20:25.150 align:middle line:84% This is a famous picture taken by the scientist here, Rosalind 00:20:25.150 --> 00:20:26.190 align:middle line:90% and Raymond. 00:20:26.190 --> 00:20:29.590 align:middle line:84% And this is essentially the first X-ray diffraction spectra 00:20:29.590 --> 00:20:32.990 align:middle line:90% of DNA. 00:20:32.990 --> 00:20:36.870 align:middle line:84% Rumor says that Watson and Crick, who actually discovered 00:20:36.870 --> 00:20:39.990 align:middle line:84% the double-helix structure of DNA, 00:20:39.990 --> 00:20:42.290 align:middle line:90% they actually saw this picture. 00:20:42.290 --> 00:20:44.590 align:middle line:84% And this is what really inspired them 00:20:44.590 --> 00:20:47.850 align:middle line:84% to propose the structural double-helix model of DNA. 00:20:47.850 --> 00:20:50.640 align:middle line:90% 00:20:50.640 --> 00:20:52.730 align:middle line:84% Now, you look at the picture here. 00:20:52.730 --> 00:20:56.870 align:middle line:84% You see this basically a cross pattern with a few discrete dots 00:20:56.870 --> 00:20:58.570 align:middle line:90% going in the two X's. 00:20:58.570 --> 00:21:01.550 align:middle line:84% This is a characteristic diffraction pattern 00:21:01.550 --> 00:21:04.670 align:middle line:90% if you have a helical structure. 00:21:04.670 --> 00:21:06.970 align:middle line:84% To prove to you that that's actually the case, 00:21:06.970 --> 00:21:08.830 align:middle line:84% we can actually do an experiment. 00:21:08.830 --> 00:21:10.630 align:middle line:90% I can show you that. 00:21:10.630 --> 00:21:13.150 align:middle line:90% I have a helical structure here. 00:21:13.150 --> 00:21:16.950 align:middle line:84% I have my light source, which is my laser pointer. 00:21:16.950 --> 00:21:18.560 align:middle line:84% I will show you that the diffraction 00:21:18.560 --> 00:21:21.560 align:middle line:84% of a laser light on this spring is 00:21:21.560 --> 00:21:22.940 align:middle line:90% going to give you that pattern. 00:21:22.940 --> 00:21:23.920 align:middle line:90% That's what I will do. 00:21:23.920 --> 00:21:27.160 align:middle line:84% But what the tricky part here is that if you 00:21:27.160 --> 00:21:30.320 align:middle line:84% look at the structure size here, of course, this is pretty big. 00:21:30.320 --> 00:21:33.560 align:middle line:84% So it means that the wavelength is actually much, much smaller 00:21:33.560 --> 00:21:36.470 align:middle line:84% than the structure feature size here. 00:21:36.470 --> 00:21:38.720 align:middle line:84% This is actually very similar to electron diffraction, 00:21:38.720 --> 00:21:40.120 align:middle line:84% which we're going to talk about later. 00:21:40.120 --> 00:21:42.162 align:middle line:84% For electron diffraction, the electron wavelength 00:21:42.162 --> 00:21:44.752 align:middle line:84% is also much smaller compared to crystalline lattice constant. 00:21:44.752 --> 00:21:46.960 align:middle line:84% So as a result, if you go through the [? secondary ?] 00:21:46.960 --> 00:21:49.200 align:middle line:84% diffraction theory, it shows you that the differential angle 00:21:49.200 --> 00:21:50.242 align:middle line:90% becomes very, very small. 00:21:50.242 --> 00:21:52.800 align:middle line:90% 00:21:52.800 --> 00:21:55.580 align:middle line:84% So that's why when I want to do this experiment, 00:21:55.580 --> 00:21:58.980 align:middle line:84% I want to do it essentially over long distances 00:21:58.980 --> 00:22:01.720 align:middle line:84% so that you can see the diffraction pattern going 00:22:01.720 --> 00:22:02.480 align:middle line:90% through here. 00:22:02.480 --> 00:22:05.300 align:middle line:84% So you may have to take a picture there, 00:22:05.300 --> 00:22:09.540 align:middle line:84% because this may be a bit difficult to align. 00:22:09.540 --> 00:22:12.080 align:middle line:84% You will have to look at it from the board there. 00:22:12.080 --> 00:22:17.160 align:middle line:90% And you can get a bit closer. 00:22:17.160 --> 00:22:19.440 align:middle line:90% Do you see anything? 00:22:19.440 --> 00:22:21.640 align:middle line:90% Do you see the cross pattern? 00:22:21.640 --> 00:22:22.800 align:middle line:90% AUDIENCE: Yes. 00:22:22.800 --> 00:22:25.640 align:middle line:84% JUEJUN HU: And it looks very similar to that, actually. 00:22:25.640 --> 00:22:27.020 align:middle line:90% Yeah. 00:22:27.020 --> 00:22:34.120 align:middle line:90% AUDIENCE: [INAUDIBLE] coming? 00:22:34.120 --> 00:22:36.800 align:middle line:84% JUEJUN HU: Yeah, my hand is not very steady. 00:22:36.800 --> 00:22:37.340 align:middle line:90% Let me see. 00:22:37.340 --> 00:22:39.100 align:middle line:90% I can see if I can try to-- 00:22:39.100 --> 00:22:43.000 align:middle line:90% 00:22:43.000 --> 00:22:44.060 align:middle line:90% is this any better? 00:22:44.060 --> 00:22:44.900 align:middle line:90% AUDIENCE: Oh, yeah. 00:22:44.900 --> 00:22:45.733 align:middle line:90% That's better, yeah. 00:22:45.733 --> 00:22:48.960 align:middle line:90% 00:22:48.960 --> 00:22:50.300 align:middle line:90% JUEJUN HU: Oh. 00:22:50.300 --> 00:22:51.340 align:middle line:90% AUDIENCE: That's cool. 00:22:51.340 --> 00:22:55.030 align:middle line:90% 00:22:55.030 --> 00:22:56.780 align:middle line:84% JUEJUN HU: Are you able to take a picture? 00:22:56.780 --> 00:22:57.405 align:middle line:90% AUDIENCE: Yeah. 00:22:57.405 --> 00:22:59.730 align:middle line:84% JUEJUN HU: You see the cross pattern? 00:22:59.730 --> 00:23:00.240 align:middle line:90% OK. 00:23:00.240 --> 00:23:01.240 align:middle line:90% Good? 00:23:01.240 --> 00:23:02.440 align:middle line:90% AUDIENCE: One more. 00:23:02.440 --> 00:23:03.440 align:middle line:90% JUEJUN HU: Oh, one more. 00:23:03.440 --> 00:23:04.720 align:middle line:90% OK. 00:23:04.720 --> 00:23:06.900 align:middle line:84% I just have to have very steady hands. 00:23:06.900 --> 00:23:24.690 align:middle line:90% 00:23:24.690 --> 00:23:25.370 align:middle line:90% Good? 00:23:25.370 --> 00:23:27.170 align:middle line:90% OK. 00:23:27.170 --> 00:23:29.762 align:middle line:90% All right. 00:23:29.762 --> 00:23:31.970 align:middle line:84% So as you'll see later, this is actually very similar 00:23:31.970 --> 00:23:34.130 align:middle line:84% to electron diffraction-- same idea. 00:23:34.130 --> 00:23:36.630 align:middle line:84% Electron diffraction also has very small diffraction angles. 00:23:36.630 --> 00:23:38.393 align:middle line:84% So essentially, they use a beam blocker 00:23:38.393 --> 00:23:40.810 align:middle line:84% to block the center beam and the diffraction pattern right 00:23:40.810 --> 00:23:42.270 align:middle line:90% around it at very small angles. 00:23:42.270 --> 00:23:44.490 align:middle line:90% So same idea. 00:23:44.490 --> 00:23:46.490 align:middle line:90% All right. 00:23:46.490 --> 00:23:50.370 align:middle line:84% Now of course, over time, people are 00:23:50.370 --> 00:23:52.773 align:middle line:84% trying to solve more and more complex structures. 00:23:52.773 --> 00:23:54.690 align:middle line:84% And then some of these complex structures, you 00:23:54.690 --> 00:23:56.990 align:middle line:84% can no longer guess it from intuition. 00:23:56.990 --> 00:23:59.450 align:middle line:84% This is just one example of actually a structure 00:23:59.450 --> 00:24:02.650 align:middle line:84% factor of some kind of protein's macromolecules. 00:24:02.650 --> 00:24:04.490 align:middle line:84% We'll talk about what structure factor is. 00:24:04.490 --> 00:24:06.930 align:middle line:84% But obviously, this is beyond any kind 00:24:06.930 --> 00:24:08.510 align:middle line:90% of a simple human intuition. 00:24:08.510 --> 00:24:11.010 align:middle line:84% And people actually have to solve it using advanced computer 00:24:11.010 --> 00:24:12.890 align:middle line:90% codes. 00:24:12.890 --> 00:24:15.650 align:middle line:84% Now coming back, we are going to focus specifically now 00:24:15.650 --> 00:24:18.130 align:middle line:84% on crystalline material, about amorphous material. 00:24:18.130 --> 00:24:20.090 align:middle line:84% So for amorphous material, what I'm showing 00:24:20.090 --> 00:24:22.850 align:middle line:84% you here is actually a simple differential structure 00:24:22.850 --> 00:24:25.090 align:middle line:90% of some kind of metallic glass. 00:24:25.090 --> 00:24:28.170 align:middle line:84% This red trace is actually a more fully amorphous metallic 00:24:28.170 --> 00:24:28.850 align:middle line:90% glass. 00:24:28.850 --> 00:24:30.890 align:middle line:84% This black trace is after annealing, which 00:24:30.890 --> 00:24:32.110 align:middle line:90% has partially crystallized. 00:24:32.110 --> 00:24:35.492 align:middle line:84% So you are all familiar with what are actually 00:24:35.492 --> 00:24:37.950 align:middle line:84% differential equations look like for crystalline materials. 00:24:37.950 --> 00:24:39.450 align:middle line:84% If you plot it essentially intensity 00:24:39.450 --> 00:24:42.850 align:middle line:84% or the counts from the detector as a function of 2 times 00:24:42.850 --> 00:24:46.050 align:middle line:84% the right angle, you see some very sharp peaks here. 00:24:46.050 --> 00:24:47.587 align:middle line:84% Each of the sharp peaks essentially 00:24:47.587 --> 00:24:49.170 align:middle line:84% corresponds to some crystalline planes 00:24:49.170 --> 00:24:51.650 align:middle line:84% with a specific spacing such as Bragg's equation. 00:24:51.650 --> 00:24:54.630 align:middle line:84% Bragg's equation is actually satisfied. 00:24:54.630 --> 00:24:56.690 align:middle line:84% And that's why you see a sharp peak there. 00:24:56.690 --> 00:24:58.590 align:middle line:84% For amorphous material, on the other hand, 00:24:58.590 --> 00:25:01.770 align:middle line:84% you tend to see this kind of broad background 00:25:01.770 --> 00:25:04.470 align:middle line:84% that looks somewhat featureless almost. 00:25:04.470 --> 00:25:08.340 align:middle line:90% 00:25:08.340 --> 00:25:13.610 align:middle line:84% So the background tends to be a lot weaker compared 00:25:13.610 --> 00:25:15.750 align:middle line:84% to these crystalline sharp peaks. 00:25:15.750 --> 00:25:18.330 align:middle line:90% 00:25:18.330 --> 00:25:21.690 align:middle line:84% This is because when you have one of these sharp peaks 00:25:21.690 --> 00:25:23.970 align:middle line:84% here, what actually happens is that you 00:25:23.970 --> 00:25:26.770 align:middle line:84% have coherent addition of the scattering from all 00:25:26.770 --> 00:25:29.668 align:middle line:84% the atoms in the material in order to generate 00:25:29.668 --> 00:25:30.710 align:middle line:90% one of those sharp peaks. 00:25:30.710 --> 00:25:33.910 align:middle line:84% Whereas for this amorphous background, 00:25:33.910 --> 00:25:35.810 align:middle line:84% you don't have that coherent addition 00:25:35.810 --> 00:25:37.890 align:middle line:90% because of irregular structure. 00:25:37.890 --> 00:25:42.130 align:middle line:84% And so that's why the intensity tends to be a lot lower there. 00:25:42.130 --> 00:25:45.290 align:middle line:84% And also, one question that will come up 00:25:45.290 --> 00:25:48.030 align:middle line:84% is, if you see this kind of spectrum, 00:25:48.030 --> 00:25:49.630 align:middle line:90% obviously, this is amorphous. 00:25:49.630 --> 00:25:52.490 align:middle line:84% But will we be able to get a bit more information out 00:25:52.490 --> 00:25:53.830 align:middle line:90% of this broad spectrum here? 00:25:53.830 --> 00:25:55.570 align:middle line:84% Can we get some quantitative information 00:25:55.570 --> 00:25:59.655 align:middle line:84% to describe the amorphous structure there? 00:25:59.655 --> 00:26:01.030 align:middle line:84% So that's what we're going to do. 00:26:01.030 --> 00:26:04.210 align:middle line:84% We're going to try to develop a quantitative theory that 00:26:04.210 --> 00:26:08.660 align:middle line:84% allows us to correlate this kind of broad, seemingly 00:26:08.660 --> 00:26:13.580 align:middle line:84% featureless spectrum to some quantitative structure 00:26:13.580 --> 00:26:17.940 align:middle line:84% descriptors of amorphous materials. 00:26:17.940 --> 00:26:20.180 align:middle line:84% So I will start out by essentially setting up 00:26:20.180 --> 00:26:22.860 align:middle line:90% the problem here. 00:26:22.860 --> 00:26:27.580 align:middle line:84% So here, basically, we will have essentially some X-rays coming 00:26:27.580 --> 00:26:30.660 align:middle line:84% into the material and then get diffracted. 00:26:30.660 --> 00:26:33.500 align:middle line:84% For any quantity that's associated with the incident 00:26:33.500 --> 00:26:37.900 align:middle line:84% X-ray, I use a subscript i for any quantity associated 00:26:37.900 --> 00:26:44.340 align:middle line:84% with the scattered or diffracted X-ray, I use the subscript s. 00:26:44.340 --> 00:26:47.180 align:middle line:84% The position of the atom, atom m, 00:26:47.180 --> 00:26:50.300 align:middle line:84% I can label atom using this letter here. 00:26:50.300 --> 00:26:52.740 align:middle line:84% So in a solid material, you can have lots of atoms here, 00:26:52.740 --> 00:26:56.420 align:middle line:84% from 1 to capital N. M is basically the label 00:26:56.420 --> 00:26:58.660 align:middle line:90% to label one each atom here OK. 00:26:58.660 --> 00:27:03.438 align:middle line:84% So r will be the position vector of that particular atom here. 00:27:03.438 --> 00:27:04.980 align:middle line:84% Now a few assumptions we want to make 00:27:04.980 --> 00:27:06.980 align:middle line:84% before we go through the quantitative derivation 00:27:06.980 --> 00:27:09.260 align:middle line:84% is that first thing, we are approximating the incident 00:27:09.260 --> 00:27:12.140 align:middle line:84% and diffracted X-ray both as monochromatic plane waves. 00:27:12.140 --> 00:27:15.100 align:middle line:84% Now monochromatic, usually that's a pretty good assumption. 00:27:15.100 --> 00:27:16.620 align:middle line:84% For X-ray diffraction, you usually 00:27:16.620 --> 00:27:19.820 align:middle line:84% have a source that's close to being monochromatic-- copper 00:27:19.820 --> 00:27:21.058 align:middle line:90% K-alpha source, for example. 00:27:21.058 --> 00:27:22.600 align:middle line:84% They usually are pretty narrow bands, 00:27:22.600 --> 00:27:24.860 align:middle line:84% so you can treat them as single wavelengths. 00:27:24.860 --> 00:27:29.460 align:middle line:84% Plane wave essentially means that your X-ray source 00:27:29.460 --> 00:27:33.100 align:middle line:84% is sufficiently far away to the sample, 00:27:33.100 --> 00:27:35.580 align:middle line:84% so that the distance of the source to a sample 00:27:35.580 --> 00:27:38.500 align:middle line:84% is much larger compared to the linear dimension of the sample. 00:27:38.500 --> 00:27:41.300 align:middle line:84% So in that case, the angular divergence of the X-ray 00:27:41.300 --> 00:27:42.640 align:middle line:90% is actually pretty small. 00:27:42.640 --> 00:27:45.760 align:middle line:84% You can treat it more or less as a plane wave. 00:27:45.760 --> 00:27:49.900 align:middle line:84% Now, next one, elastic scattering, 00:27:49.900 --> 00:27:51.860 align:middle line:84% means that the scatter in the incident X-ray 00:27:51.860 --> 00:27:54.723 align:middle line:84% has exactly the same energy or same wavelength. 00:27:54.723 --> 00:27:56.140 align:middle line:84% This essentially means that we are 00:27:56.140 --> 00:27:59.200 align:middle line:84% discounting any of the Compton scattering effect. 00:27:59.200 --> 00:28:01.820 align:middle line:84% Compton scattering is one very common elastic scattering 00:28:01.820 --> 00:28:06.060 align:middle line:84% mechanism where actually, the X-ray wavelength changes. 00:28:06.060 --> 00:28:08.700 align:middle line:84% Last one, we are neglecting any X-ray attenuation 00:28:08.700 --> 00:28:10.680 align:middle line:84% in the sample, which obviously is incorrect. 00:28:10.680 --> 00:28:13.340 align:middle line:84% Because when the X-ray gets scattered, 00:28:13.340 --> 00:28:16.340 align:middle line:84% the initial beam must have some energy getting diverted 00:28:16.340 --> 00:28:18.220 align:middle line:90% into the scattered X-ray. 00:28:18.220 --> 00:28:20.860 align:middle line:84% What we are assuming there is that the effect 00:28:20.860 --> 00:28:22.460 align:middle line:90% is relatively weak. 00:28:22.460 --> 00:28:27.300 align:middle line:84% So essentially, the field amplitude for the incident X-ray 00:28:27.300 --> 00:28:30.360 align:middle line:84% is uniform across the entire sample for all the atoms. 00:28:30.360 --> 00:28:32.420 align:middle line:84% That's what we're basically assuming. 00:28:32.420 --> 00:28:37.420 align:middle line:84% It is a reasonable assumption if the sample is not too thick. 00:28:37.420 --> 00:28:40.100 align:middle line:84% All right, now let's start to take a look at how we actually 00:28:40.100 --> 00:28:41.540 align:middle line:90% set up this problem here. 00:28:41.540 --> 00:28:43.380 align:middle line:84% So we will start out by essentially 00:28:43.380 --> 00:28:46.460 align:middle line:84% trying to write the complex amplitude of the incident 00:28:46.460 --> 00:28:50.598 align:middle line:84% wave, which is basically this equation here. 00:28:50.598 --> 00:28:52.140 align:middle line:84% I assume you're all familiar with how 00:28:52.140 --> 00:28:54.015 align:middle line:84% to represent the waves in a complex amplitude 00:28:54.015 --> 00:28:54.800 align:middle line:90% representation. 00:28:54.800 --> 00:28:56.580 align:middle line:84% So you will have essentially the amplitude 00:28:56.580 --> 00:29:01.220 align:middle line:84% E. This is basically the real amplitude of the incident X-ray. 00:29:01.220 --> 00:29:05.110 align:middle line:84% And this exponential factor then gives you the phase delay 00:29:05.110 --> 00:29:07.550 align:middle line:84% for light to propagate from the origin point all 00:29:07.550 --> 00:29:10.430 align:middle line:84% the way to this point r here with some kind of position 00:29:10.430 --> 00:29:11.438 align:middle line:90% vector. 00:29:11.438 --> 00:29:13.230 align:middle line:84% So if you're looking at the field amplitude 00:29:13.230 --> 00:29:15.630 align:middle line:84% specifically at one particular atom m, 00:29:15.630 --> 00:29:18.810 align:middle line:84% then you just plug in essentially rm into r here. 00:29:18.810 --> 00:29:20.390 align:middle line:84% And this is basically field amplitude 00:29:20.390 --> 00:29:23.350 align:middle line:90% at that particular atom. 00:29:23.350 --> 00:29:25.790 align:middle line:84% So now the next thing we argue here 00:29:25.790 --> 00:29:31.470 align:middle line:84% is m when we are looking at this scattered field amplitude Es, 00:29:31.470 --> 00:29:34.470 align:middle line:84% we will argue that, first thing, this scattered field amplitude, 00:29:34.470 --> 00:29:38.313 align:middle line:84% it must be proportional to the incident field amplitude. 00:29:38.313 --> 00:29:39.230 align:middle line:90% This also makes sense. 00:29:39.230 --> 00:29:41.210 align:middle line:84% For any kind of linear scattering phenomena, 00:29:41.210 --> 00:29:43.690 align:middle line:84% the more photons coming from the incident light, 00:29:43.690 --> 00:29:47.190 align:middle line:84% the more it will scatter proportionally. 00:29:47.190 --> 00:29:54.310 align:middle line:84% So therefore, we postulate that this Es should scale with Ei. 00:29:54.310 --> 00:29:56.630 align:middle line:84% And therefore, if we plug in Ei, it essentially 00:29:56.630 --> 00:30:03.990 align:middle line:84% just means that E negative ikirm, 00:30:03.990 --> 00:30:10.190 align:middle line:84% this will be the scattered field amplitude by the atom m. 00:30:10.190 --> 00:30:11.570 align:middle line:90% I did not use negative here. 00:30:11.570 --> 00:30:14.750 align:middle line:84% I used the positive, so this is just-- 00:30:14.750 --> 00:30:16.430 align:middle line:90% let me make this positive. 00:30:16.430 --> 00:30:20.070 align:middle line:90% 00:30:20.070 --> 00:30:21.970 align:middle line:84% So this is basically the incident field. 00:30:21.970 --> 00:30:25.950 align:middle line:90% This is the excitation source. 00:30:25.950 --> 00:30:28.710 align:middle line:84% Now, you also need to have a second term here because you are 00:30:28.710 --> 00:30:30.330 align:middle line:90% measuring the scattering field. 00:30:30.330 --> 00:30:32.430 align:middle line:84% The scattering field, as it propagates away 00:30:32.430 --> 00:30:34.270 align:middle line:84% from this atom to your detector, it's 00:30:34.270 --> 00:30:36.530 align:middle line:84% also going to accumulate a phase delay as well. 00:30:36.530 --> 00:30:46.910 align:middle line:84% So the phase delay term will scale with the iks r minus rm. 00:30:46.910 --> 00:30:50.750 align:middle line:84% So the last term here, r minus rm, 00:30:50.750 --> 00:30:56.630 align:middle line:84% is basically the distance the light will travel from your atom 00:30:56.630 --> 00:30:58.950 align:middle line:84% to where your detector is or where you are 00:30:58.950 --> 00:31:01.430 align:middle line:90% measuring the scattering field. 00:31:01.430 --> 00:31:12.550 align:middle line:90% 00:31:12.550 --> 00:31:17.430 align:middle line:84% So the next thing we do, we notice that both of these 00:31:17.430 --> 00:31:21.390 align:middle line:84% contains essentially rm here a complex dimension containing rm. 00:31:21.390 --> 00:31:24.030 align:middle line:84% So we can regroup the terms a little bit. 00:31:24.030 --> 00:31:27.670 align:middle line:90% We can have E, now have iks r. 00:31:27.670 --> 00:31:30.430 align:middle line:90% 00:31:30.430 --> 00:31:35.270 align:middle line:84% And then the second term, we have E negative iki. 00:31:35.270 --> 00:31:37.990 align:middle line:90% 00:31:37.990 --> 00:31:41.630 align:middle line:84% Let's pull out this common term of rm here. 00:31:41.630 --> 00:31:44.710 align:middle line:84% And then you now have ks minus ki. 00:31:44.710 --> 00:31:54.950 align:middle line:90% 00:31:54.950 --> 00:31:57.680 align:middle line:84% I'm just pulling out a common term that contains 00:31:57.680 --> 00:31:59.420 align:middle line:90% this rm, the complex measure. 00:31:59.420 --> 00:32:02.800 align:middle line:90% 00:32:02.800 --> 00:32:06.560 align:middle line:84% Now, we can also define this new quantity here 00:32:06.560 --> 00:32:14.720 align:middle line:84% that we call Q, which is what we call the scattering vector. 00:32:14.720 --> 00:32:21.940 align:middle line:84% Q should equal to is minus ks, I should say. 00:32:21.940 --> 00:32:25.120 align:middle line:90% 00:32:25.120 --> 00:32:29.920 align:middle line:84% This is the difference of the wave vector for the scattered 00:32:29.920 --> 00:32:32.920 align:middle line:90% and the incident X-ray waves. 00:32:32.920 --> 00:32:36.600 align:middle line:90% 00:32:36.600 --> 00:32:39.120 align:middle line:84% So if you do that, the last vector 00:32:39.120 --> 00:32:43.160 align:middle line:84% just becomes an exponential negative iQrm. 00:32:43.160 --> 00:32:50.640 align:middle line:90% 00:32:50.640 --> 00:32:52.360 align:middle line:84% You'll also notice that in this equation, 00:32:52.360 --> 00:32:54.000 align:middle line:90% I did not use equals sign. 00:32:54.000 --> 00:32:55.840 align:middle line:90% I use a proportional sign-- 00:32:55.840 --> 00:32:57.120 align:middle line:90% proportionality sign. 00:32:57.120 --> 00:32:59.520 align:middle line:84% And that's because we actually don't 00:32:59.520 --> 00:33:04.360 align:middle line:84% know what is the scattering cross-section for the atom m. 00:33:04.360 --> 00:33:09.680 align:middle line:84% So what we can do in order to eliminate this uncertainty here, 00:33:09.680 --> 00:33:13.920 align:middle line:84% we can lump all of the essential atomic scattering cross-section 00:33:13.920 --> 00:33:17.520 align:middle line:84% all into one constant layer that we call the fm. 00:33:17.520 --> 00:33:21.120 align:middle line:90% 00:33:21.120 --> 00:33:26.120 align:middle line:84% Essentially, we are lumping the scattering cross-section, 00:33:26.120 --> 00:33:29.960 align:middle line:84% scattering efficiency, as well as this Ee, electric field, 00:33:29.960 --> 00:33:32.620 align:middle line:84% and in addition, also this factor here. 00:33:32.620 --> 00:33:35.520 align:middle line:84% So this factor is actually a constant 00:33:35.520 --> 00:33:37.425 align:middle line:84% if you have a fixed detector partition. 00:33:37.425 --> 00:33:38.800 align:middle line:84% R is basically where you actually 00:33:38.800 --> 00:33:40.035 align:middle line:90% measure the scatter field. 00:33:40.035 --> 00:33:41.660 align:middle line:84% So you have a fixed detector partition, 00:33:41.660 --> 00:33:42.920 align:middle line:84% this is basically a constant there. 00:33:42.920 --> 00:33:44.837 align:middle line:84% So we are lumping all these constants together 00:33:44.837 --> 00:33:46.760 align:middle line:90% into one single constant. 00:33:46.760 --> 00:33:54.040 align:middle line:84% This is what we call the scattering efficiency 00:33:54.040 --> 00:33:57.892 align:middle line:90% or scattering factor. 00:33:57.892 --> 00:34:01.360 align:middle line:84% So scaling factor is basically about how efficient 00:34:01.360 --> 00:34:04.240 align:middle line:84% this atom can use scatter light, scatter X-rays. 00:34:04.240 --> 00:34:09.880 align:middle line:90% 00:34:09.880 --> 00:34:13.159 align:middle line:84% All right, so now, next thing we can do, 00:34:13.159 --> 00:34:18.199 align:middle line:90% we want to try to add up this. 00:34:18.199 --> 00:34:22.320 align:middle line:84% This gives you the full amplitude 00:34:22.320 --> 00:34:27.040 align:middle line:84% that's being scattered by one single atom m. 00:34:27.040 --> 00:34:29.679 align:middle line:84% In a solid material, you have a group 00:34:29.679 --> 00:34:34.800 align:middle line:84% of atoms, a total of N number of atoms-- capital N. 00:34:34.800 --> 00:34:38.120 align:middle line:84% So therefore, we expect that in order 00:34:38.120 --> 00:34:43.440 align:middle line:84% to get the total scattered amplitude 00:34:43.440 --> 00:34:44.780 align:middle line:90% you need to add everything up. 00:34:44.780 --> 00:34:48.159 align:middle line:90% So the total amplitude-- 00:34:48.159 --> 00:34:50.040 align:middle line:90% let's see, what letter do I use. 00:34:50.040 --> 00:34:53.469 align:middle line:84% The total amplitude, let's write it as E total. 00:34:53.469 --> 00:34:56.570 align:middle line:90% 00:34:56.570 --> 00:35:01.650 align:middle line:84% Basically, we should sum over all the possible Es. 00:35:01.650 --> 00:35:05.050 align:middle line:90% 00:35:05.050 --> 00:35:08.010 align:middle line:84% So summing over all the capital N 00:35:08.010 --> 00:35:11.870 align:middle line:84% number of atoms in the solid material, all of the amplitudes 00:35:11.870 --> 00:35:12.370 align:middle line:90% here. 00:35:12.370 --> 00:35:14.250 align:middle line:84% They add up coherently because we assume 00:35:14.250 --> 00:35:15.890 align:middle line:90% this is a monochromatic wave. 00:35:15.890 --> 00:35:19.710 align:middle line:84% And therefore, instead of adding up the intensity, 00:35:19.710 --> 00:35:22.170 align:middle line:84% which adds up the amplitude because it's 00:35:22.170 --> 00:35:23.290 align:middle line:90% a single wavelength. 00:35:23.290 --> 00:35:25.930 align:middle line:84% If it's all single wavelength, then it's always coherent. 00:35:25.930 --> 00:35:27.850 align:middle line:90% So it adds up the amplitude. 00:35:27.850 --> 00:35:29.530 align:middle line:90% So plug in there. 00:35:29.530 --> 00:35:31.410 align:middle line:84% That essentially means that you're summing up 00:35:31.410 --> 00:35:36.550 align:middle line:90% m n to all the fm e iQrm. 00:35:36.550 --> 00:35:43.900 align:middle line:90% 00:35:43.900 --> 00:35:47.370 align:middle line:84% Now, this is the field amplitude. 00:35:47.370 --> 00:35:49.750 align:middle line:84% What you measure is actually the intensity. 00:35:49.750 --> 00:35:52.570 align:middle line:84% You actually measure [INAUDIBLE] intensity, not amplitude. 00:35:52.570 --> 00:35:56.210 align:middle line:84% So intensity actually scales with amplitude squared. 00:35:56.210 --> 00:36:00.330 align:middle line:84% So if you want to measure the scatter intensity I, 00:36:00.330 --> 00:36:07.330 align:middle line:84% then what you need to do is take basically the amplitude modulus 00:36:07.330 --> 00:36:08.070 align:middle line:90% squared. 00:36:08.070 --> 00:36:12.562 align:middle line:90% 00:36:12.562 --> 00:36:16.050 align:middle line:84% So how do you do a modulus squared for a complex number 00:36:16.050 --> 00:36:18.610 align:middle line:90% here? 00:36:18.610 --> 00:36:23.290 align:middle line:84% A modulus squared essentially is equal to the real part 00:36:23.290 --> 00:36:26.580 align:middle line:90% times its imaginary part. 00:36:26.580 --> 00:36:28.830 align:middle line:84% That's how you do modulus squared for complex numbers. 00:36:28.830 --> 00:36:31.650 align:middle line:84% So therefore, that means that you now 00:36:31.650 --> 00:36:35.290 align:middle line:84% have essentially this one, the real part, which I just 00:36:35.290 --> 00:36:44.450 align:middle line:84% copy this E negative iQrm then times the imaginary part. 00:36:44.450 --> 00:36:49.420 align:middle line:84% The imaginary part, I use a different letter n here-- 00:36:49.420 --> 00:36:56.080 align:middle line:90% lowercase n, e now complex. 00:36:56.080 --> 00:37:00.825 align:middle line:90% 00:37:00.825 --> 00:37:02.700 align:middle line:84% Essentially, you get rid of the negative sign 00:37:02.700 --> 00:37:04.640 align:middle line:84% there because that's a complex conjugate. 00:37:04.640 --> 00:37:07.460 align:middle line:84% And I use a different letter n here 00:37:07.460 --> 00:37:10.460 align:middle line:84% to indicate that in each of the parentheses, 00:37:10.460 --> 00:37:14.580 align:middle line:84% we have a total of uppercase N number of terms. 00:37:14.580 --> 00:37:17.880 align:middle line:84% So we have an n terms times also an n terms here. 00:37:17.880 --> 00:37:20.440 align:middle line:84% So you actually basically get n squared terms in this product 00:37:20.440 --> 00:37:20.940 align:middle line:90% here. 00:37:20.940 --> 00:37:29.027 align:middle line:90% 00:37:29.027 --> 00:37:30.860 align:middle line:84% AUDIENCE: What was the significance of the n 00:37:30.860 --> 00:37:32.060 align:middle line:90% squared terms? 00:37:32.060 --> 00:37:34.892 align:middle line:90% I understand that it's true. 00:37:34.892 --> 00:37:37.580 align:middle line:84% JUEJUN HU: I'm just telling you that [INAUDIBLE]. 00:37:37.580 --> 00:37:38.160 align:middle line:90% That's it. 00:37:38.160 --> 00:37:39.420 align:middle line:90% [CHUCKLES] 00:37:39.420 --> 00:37:41.920 align:middle line:84% AUDIENCE: But what was that to lowercase n? 00:37:41.920 --> 00:37:44.300 align:middle line:84% JUEJUN HU: Lowercase n is just another way, 00:37:44.300 --> 00:37:46.740 align:middle line:84% I can replace it with any letter there. 00:37:46.740 --> 00:37:49.380 align:middle line:84% It's basically, this subscript has 00:37:49.380 --> 00:37:51.900 align:middle line:84% to cycle from 1 all the way to capital n. 00:37:51.900 --> 00:37:54.780 align:middle line:84% You have to sum over all the atoms-- basically independent. 00:37:54.780 --> 00:37:55.800 align:middle line:84% That's what I'm trying to indicate. 00:37:55.800 --> 00:37:56.883 align:middle line:90% Yeah, they're independent. 00:37:56.883 --> 00:38:00.980 align:middle line:90% 00:38:00.980 --> 00:38:04.215 align:middle line:84% OK, so the whole problem of solving the diffraction 00:38:04.215 --> 00:38:06.840 align:middle line:84% intensity essentially boils down to solving this equation here. 00:38:06.840 --> 00:38:09.900 align:middle line:90% 00:38:09.900 --> 00:38:12.340 align:middle line:84% Also, before I move any further, I 00:38:12.340 --> 00:38:14.700 align:middle line:84% want to introduce this additional parameter here. 00:38:14.700 --> 00:38:17.740 align:middle line:84% This is called SQ This is called structure factor. 00:38:17.740 --> 00:38:21.900 align:middle line:84% By definition, we are saying that this intensity equals 00:38:21.900 --> 00:38:36.180 align:middle line:90% to SQ times N times fm squared. 00:38:36.180 --> 00:38:41.300 align:middle line:84% So here, this bracket indicates averaging. 00:38:41.300 --> 00:38:45.860 align:middle line:84% So bracket fm, essentially, this is the average scatter factor 00:38:45.860 --> 00:38:49.020 align:middle line:84% across the entire solid, across all the atoms. 00:38:49.020 --> 00:38:55.220 align:middle line:84% So this is the definition of SQ the structure factor here, 00:38:55.220 --> 00:38:56.620 align:middle line:90% just the definition. 00:38:56.620 --> 00:38:58.300 align:middle line:90% That's it. 00:38:58.300 --> 00:39:02.940 align:middle line:84% So what we are seeing here is basically the structure factor. 00:39:02.940 --> 00:39:06.660 align:middle line:84% This is directly correlated with the measured diffraction 00:39:06.660 --> 00:39:07.360 align:middle line:90% intensity. 00:39:07.360 --> 00:39:08.940 align:middle line:84% So if you know differential intensity 00:39:08.940 --> 00:39:11.180 align:middle line:84% you can immediately calculate structure factor. 00:39:11.180 --> 00:39:15.420 align:middle line:84% Structure factor is also a function of Q. So what is Q? 00:39:15.420 --> 00:39:16.980 align:middle line:90% Let's now take a look at Q here. 00:39:16.980 --> 00:39:22.620 align:middle line:90% Q is ks minus ki 00:39:22.620 --> 00:39:28.060 align:middle line:84% So suppose this is the solid surface. 00:39:28.060 --> 00:39:31.060 align:middle line:84% In a crystalline material, this may be one of the atomic planes. 00:39:31.060 --> 00:39:34.860 align:middle line:84% Then you have an X-ray comes in here. 00:39:34.860 --> 00:39:38.220 align:middle line:84% This is the X-ray diffraction angle theta. 00:39:38.220 --> 00:39:40.740 align:middle line:90% This is incident X-ray. 00:39:40.740 --> 00:39:44.610 align:middle line:84% It will then diffract let's say go to a similar [INAUDIBLE]. 00:39:44.610 --> 00:39:47.900 align:middle line:90% This is also theta. 00:39:47.900 --> 00:39:50.030 align:middle line:84% So in this configuration here, you 00:39:50.030 --> 00:39:53.510 align:middle line:84% can now plot essentially using simple vector identity 00:39:53.510 --> 00:39:56.630 align:middle line:90% to see what is the ks and ki. 00:39:56.630 --> 00:40:02.310 align:middle line:84% So suppose this is a ki, this is ks. 00:40:02.310 --> 00:40:06.230 align:middle line:84% Then it's straightforward to see that the angle here 00:40:06.230 --> 00:40:09.830 align:middle line:90% should be 2 times theta. 00:40:09.830 --> 00:40:13.510 align:middle line:84% So now ks minus ki will be the vector that 00:40:13.510 --> 00:40:15.710 align:middle line:90% links their endpoints here. 00:40:15.710 --> 00:40:18.510 align:middle line:84% And it's pretty straightforward to see that in this case, 00:40:18.510 --> 00:40:21.790 align:middle line:84% this Q, which is basically ks minus ki 00:40:21.790 --> 00:40:25.430 align:middle line:84% is equal to-- in terms of magnitude 00:40:25.430 --> 00:40:32.175 align:middle line:84% should be 2 times the magnitude of ki or ks. 00:40:32.175 --> 00:40:33.550 align:middle line:84% In this case, the magnitude of ki 00:40:33.550 --> 00:40:36.990 align:middle line:84% equals to ks because this is the elastic scattering. 00:40:36.990 --> 00:40:38.830 align:middle line:84% So they should have the same wavelength, 00:40:38.830 --> 00:40:42.350 align:middle line:90% then times sine theta. 00:40:42.350 --> 00:40:45.070 align:middle line:90% 00:40:45.070 --> 00:40:49.310 align:middle line:84% Because the half length here, this is basically a ki times 00:40:49.310 --> 00:40:50.110 align:middle line:90% sine theta. 00:40:50.110 --> 00:40:52.670 align:middle line:90% It's 2 times ki sine theta. 00:40:52.670 --> 00:41:10.510 align:middle line:90% So that's the magnitude of Q. 00:41:10.510 --> 00:41:12.710 align:middle line:84% So what this equation means is that if you 00:41:12.710 --> 00:41:14.510 align:middle line:84% are looking in isotropic solids, which 00:41:14.510 --> 00:41:17.430 align:middle line:84% means that the orientation of Q doesn't really matter, 00:41:17.430 --> 00:41:24.270 align:middle line:84% then Q is basically synonymous with theta. 00:41:24.270 --> 00:41:26.670 align:middle line:84% So what we are saying here is that the structure factor 00:41:26.670 --> 00:41:28.590 align:middle line:84% is a function of Q, essentially meaning 00:41:28.590 --> 00:41:31.230 align:middle line:90% that it's a function of theta. 00:41:31.230 --> 00:41:33.630 align:middle line:84% So if you measure I, the scattered intensity, 00:41:33.630 --> 00:41:35.950 align:middle line:84% as a function of a Bragg angle theta, 00:41:35.950 --> 00:41:40.630 align:middle line:84% then you can immediately get SQ if you know the scattering 00:41:40.630 --> 00:41:41.810 align:middle line:90% vector for all the atoms. 00:41:41.810 --> 00:41:49.932 align:middle line:90% 00:41:49.932 --> 00:41:51.890 align:middle line:84% All right, so now this comes back a little bit. 00:41:51.890 --> 00:41:56.888 align:middle line:84% Let's think about-- before we talk about amorphous materials, 00:41:56.888 --> 00:41:58.430 align:middle line:84% let's actually step back a little bit 00:41:58.430 --> 00:42:02.590 align:middle line:84% and talk about what about crystalline materials. 00:42:02.590 --> 00:42:05.150 align:middle line:84% What if it's a crystalline material? 00:42:05.150 --> 00:42:09.630 align:middle line:84% So in a crystalline material, you have a periodic lattice. 00:42:09.630 --> 00:42:11.750 align:middle line:84% If you want to have a diffraction peak, 00:42:11.750 --> 00:42:15.190 align:middle line:84% from what you learned in 3.010 or whatever, 00:42:15.190 --> 00:42:18.527 align:middle line:84% you know that need to ensure that the X-ray scattered 00:42:18.527 --> 00:42:21.110 align:middle line:84% by the different atoms or from different unit cells should add 00:42:21.110 --> 00:42:21.890 align:middle line:90% up coherently. 00:42:21.890 --> 00:42:24.590 align:middle line:84% That's how you actually get X-ray diffraction peak. 00:42:24.590 --> 00:42:27.657 align:middle line:84% So now if you look at this equation here, for example, 00:42:27.657 --> 00:42:28.990 align:middle line:90% let's look at this one, E total. 00:42:28.990 --> 00:42:32.730 align:middle line:84% In order to have coherent addition, 00:42:32.730 --> 00:42:35.390 align:middle line:84% you need to make sure that the X-ray scattered 00:42:35.390 --> 00:42:40.320 align:middle line:84% by all the different atoms, they add up in phase. 00:42:40.320 --> 00:42:43.560 align:middle line:84% So what that means is that the phase vector here, which 00:42:43.560 --> 00:42:49.680 align:middle line:84% is dictated by Q times rm they should all either be the same 00:42:49.680 --> 00:42:54.080 align:middle line:84% or differ by integer multiple of 2 pi. 00:42:54.080 --> 00:43:04.080 align:middle line:84% So effectively, what I'm saying here is that Q rm, or more 00:43:04.080 --> 00:43:06.280 align:middle line:84% rigorously I minus rm, essentially 00:43:06.280 --> 00:43:07.920 align:middle line:84% the difference of any of the two, 00:43:07.920 --> 00:43:13.890 align:middle line:84% is always equal to an integer multiple of 2 pi. 00:43:13.890 --> 00:43:17.480 align:middle line:90% M is just some integer. 00:43:17.480 --> 00:43:20.240 align:middle line:84% I neglect the rm here because you can always 00:43:20.240 --> 00:43:23.942 align:middle line:84% set the origin-- coincide origin with one of the atoms. 00:43:23.942 --> 00:43:26.400 align:middle line:84% In that case, we can essentially drop the second term here. 00:43:26.400 --> 00:43:29.020 align:middle line:90% This is fine. 00:43:29.020 --> 00:43:31.760 align:middle line:84% So now let's take a look at what this condition will lead to. 00:43:31.760 --> 00:43:35.240 align:middle line:84% So let's first essentially write what Q is. 00:43:35.240 --> 00:43:41.960 align:middle line:84% Q from there is basically a 2 times ki. 00:43:41.960 --> 00:43:43.720 align:middle line:90% What's ki? 00:43:43.720 --> 00:43:45.720 align:middle line:90% K is 2 pi divided by lambda. 00:43:45.720 --> 00:43:51.298 align:middle line:84% So it's 2 pi divided by lambda, where lambda is the wavelength. 00:43:51.298 --> 00:43:52.840 align:middle line:84% You also have some set of sine theta. 00:43:52.840 --> 00:43:56.320 align:middle line:90% 00:43:56.320 --> 00:44:05.120 align:middle line:84% Now times rm-- for rm, if you are looking 00:44:05.120 --> 00:44:08.258 align:middle line:90% at, say, two atomic planes. 00:44:08.258 --> 00:44:09.800 align:middle line:84% You're looking at scattering from two 00:44:09.800 --> 00:44:11.280 align:middle line:90% neighboring atomic planes. 00:44:11.280 --> 00:44:14.080 align:middle line:84% Then I minus rm essentially gives you 00:44:14.080 --> 00:44:17.920 align:middle line:84% the atomic spacing or the spacing between atomic planes. 00:44:17.920 --> 00:44:21.080 align:middle line:84% So if we want to be consistent with the terminology 00:44:21.080 --> 00:44:25.820 align:middle line:84% or the symbols that you use for the Bragg differential equation, 00:44:25.820 --> 00:44:29.680 align:middle line:84% essentially rm becomes d, the interatomic spacing, 00:44:29.680 --> 00:44:32.560 align:middle line:84% or the atomic plane spacing there. 00:44:32.560 --> 00:44:37.600 align:middle line:84% Then on the other side, you can take 00:44:37.600 --> 00:44:39.500 align:middle line:90% M the integer as just unity. 00:44:39.500 --> 00:44:43.035 align:middle line:90% 00:44:43.035 --> 00:44:45.000 align:middle line:90% So now let's see what we have. 00:44:45.000 --> 00:44:47.880 align:middle line:90% 2 pi, 2 pi cancels out. 00:44:47.880 --> 00:44:57.920 align:middle line:84% It basically tells you that lambda equals to 2D sine theta, 00:44:57.920 --> 00:45:01.800 align:middle line:84% which is the Bragg equation that you must have seen in previous 00:45:01.800 --> 00:45:02.460 align:middle line:90% lectures. 00:45:02.460 --> 00:45:10.640 align:middle line:90% 00:45:10.640 --> 00:45:13.160 align:middle line:84% The other way to look at it is just to look at this equation 00:45:13.160 --> 00:45:15.020 align:middle line:90% itself. 00:45:15.020 --> 00:45:19.500 align:middle line:84% So rm, this is a vector in real space. 00:45:19.500 --> 00:45:22.440 align:middle line:84% It's basically the position vector for some atoms. 00:45:22.440 --> 00:45:26.800 align:middle line:84% Q-- because 2 pi is dimensionless, 00:45:26.800 --> 00:45:29.620 align:middle line:84% so it means that the unit of Q must have 00:45:29.620 --> 00:45:32.570 align:middle line:90% [INAUDIBLE] of inverse lengths. 00:45:32.570 --> 00:45:36.430 align:middle line:84% So Q is actually defined in the reciprocal space. 00:45:36.430 --> 00:45:39.550 align:middle line:90% 00:45:39.550 --> 00:45:44.890 align:middle line:84% So a reciprocal space vector times a real space lattice 00:45:44.890 --> 00:45:47.190 align:middle line:84% vector equals to integer multiple of 2 pi. 00:45:47.190 --> 00:45:52.690 align:middle line:84% This is actually by definition a reciprocal lattice. 00:45:52.690 --> 00:45:56.930 align:middle line:84% So it also tells you that you only get a diffraction peak when 00:45:56.930 --> 00:46:00.850 align:middle line:84% the scattering vector is one of the reciprocal lattice 00:46:00.850 --> 00:46:03.450 align:middle line:84% vectors, which again is probably a conclusion you have learned 00:46:03.450 --> 00:46:08.350 align:middle line:84% in your previous lectures, I suppose. 00:46:08.350 --> 00:46:08.850 align:middle line:90% Yes? 00:46:08.850 --> 00:46:09.850 align:middle line:90% [CHUCKLES] 00:46:09.850 --> 00:46:12.035 align:middle line:90% Well, now you see it, at least. 00:46:12.035 --> 00:46:13.410 align:middle line:84% In a crystalline material, if you 00:46:13.410 --> 00:46:16.170 align:middle line:84% want to have a diffraction peak, then the diffraction vector 00:46:16.170 --> 00:46:19.535 align:middle line:84% should be one of the reciprocal lattice vectors. 00:46:19.535 --> 00:46:20.910 align:middle line:84% Especially this is by definition. 00:46:20.910 --> 00:46:23.330 align:middle line:84% This is the definition of reciprocal lattice vector. 00:46:23.330 --> 00:46:26.728 align:middle line:84% Or you can always go with Bragg equation. 00:46:26.728 --> 00:46:28.270 align:middle line:84% That's a simple way of looking at it. 00:46:28.270 --> 00:46:32.530 align:middle line:90% 00:46:32.530 --> 00:46:34.490 align:middle line:84% All right, that's for crystalline material. 00:46:34.490 --> 00:46:37.890 align:middle line:84% Now, what about for amorphous material? 00:46:37.890 --> 00:46:42.210 align:middle line:84% So for amorphous material, this is probably 00:46:42.210 --> 00:46:44.450 align:middle line:84% one of the most mathematically involved derivations 00:46:44.450 --> 00:46:46.010 align:middle line:90% throughout the whole semester. 00:46:46.010 --> 00:46:50.470 align:middle line:84% So I need to wipe out the board just to get enough space. 00:46:50.470 --> 00:46:51.970 align:middle line:84% I'm going to keep it here because we 00:46:51.970 --> 00:46:54.570 align:middle line:90% are going to try to derive SQ. 00:46:54.570 --> 00:46:56.830 align:middle line:90% But then I need all the space. 00:46:56.830 --> 00:47:10.655 align:middle line:90% 00:47:10.655 --> 00:47:13.210 align:middle line:84% AUDIENCE: Wait, so the Bragg equation came 00:47:13.210 --> 00:47:15.385 align:middle line:90% from also derivation of SQ? 00:47:15.385 --> 00:47:16.010 align:middle line:90% JUEJUN HU: Yeah 00:47:16.010 --> 00:47:16.970 align:middle line:90% AUDIENCE: OK. 00:47:16.970 --> 00:47:18.850 align:middle line:84% JUEJUN HU: Essentially, we just assume 00:47:18.850 --> 00:47:22.390 align:middle line:84% that the scatter X-ray has to add up coherently. 00:47:22.390 --> 00:47:24.278 align:middle line:84% They have to be in-phase for all the atoms. 00:47:24.278 --> 00:47:26.570 align:middle line:84% From that, we actually come up with the Bragg equation. 00:47:26.570 --> 00:47:28.610 align:middle line:84% That's the only assumption we make. 00:47:28.610 --> 00:47:29.930 align:middle line:90% Yeah. 00:47:29.930 --> 00:47:35.510 align:middle line:84% All right, so now before I talk about the amorphous, 00:47:35.510 --> 00:47:38.250 align:middle line:84% how to derive it, I do want to revisit one topic there 00:47:38.250 --> 00:47:39.910 align:middle line:84% about pair distribution function. 00:47:39.910 --> 00:47:41.827 align:middle line:84% Because what we'll see later is that actually, 00:47:41.827 --> 00:47:44.090 align:middle line:84% differential function of SQ can be directly correlated 00:47:44.090 --> 00:47:45.423 align:middle line:90% with pair distribution function. 00:47:45.423 --> 00:47:48.010 align:middle line:84% So this is a slide quoted from the previous lecture. 00:47:48.010 --> 00:47:50.930 align:middle line:84% So pair distribution function, judging from its name, 00:47:50.930 --> 00:47:55.810 align:middle line:84% is a function that measures the correlation 00:47:55.810 --> 00:47:59.530 align:middle line:84% of atomic pairs of certain spacing in between them. 00:47:59.530 --> 00:48:03.410 align:middle line:84% So in our previous slide, we have defined it essentially as-- 00:48:03.410 --> 00:48:06.890 align:middle line:84% if you put one of the atoms at the center at the origin. 00:48:06.890 --> 00:48:10.650 align:middle line:84% And then it tells you the probability 00:48:10.650 --> 00:48:13.530 align:middle line:84% of finding another atom at the spacing r 00:48:13.530 --> 00:48:15.710 align:middle line:90% away from the center atom. 00:48:15.710 --> 00:48:18.070 align:middle line:84% We also said that we can normalize it. 00:48:18.070 --> 00:48:20.490 align:middle line:84% Essentially, the number density of atoms at distance r 00:48:20.490 --> 00:48:23.950 align:middle line:84% is given by rho times gr where gr is our pair distribution 00:48:23.950 --> 00:48:25.820 align:middle line:84% function, and rho is the average number 00:48:25.820 --> 00:48:27.820 align:middle line:90% density across entire solid. 00:48:27.820 --> 00:48:29.580 align:middle line:84% If you do this kind of definition here, 00:48:29.580 --> 00:48:31.580 align:middle line:84% go with this kind of definition, then 00:48:31.580 --> 00:48:34.340 align:middle line:84% this gr should have these properties here. 00:48:34.340 --> 00:48:37.460 align:middle line:84% If the inter-atomic spacing goes to 0, 00:48:37.460 --> 00:48:41.460 align:middle line:84% g should go to 0 because no two atoms can occupy exactly 00:48:41.460 --> 00:48:45.220 align:middle line:84% the same location as lattice due to Pauli exclusion principle. 00:48:45.220 --> 00:48:48.900 align:middle line:84% Also, if r goes to infinity, meaning that if the two 00:48:48.900 --> 00:48:51.380 align:middle line:84% atoms are very, very far away, then you 00:48:51.380 --> 00:48:54.700 align:middle line:84% lose any kind of short-range or medium-range order. 00:48:54.700 --> 00:48:56.780 align:middle line:84% And this long-range disorder means 00:48:56.780 --> 00:49:00.340 align:middle line:84% that g must go to unity because you are seeing essentially 00:49:00.340 --> 00:49:03.540 align:middle line:84% a uniform loss matrix if you are very far away from the center. 00:49:03.540 --> 00:49:06.940 align:middle line:84% All kinds of near-neighbor information 00:49:06.940 --> 00:49:11.100 align:middle line:84% gets completely swept out due to the lack of long-range order. 00:49:11.100 --> 00:49:13.820 align:middle line:84% OK, so that's what we talked about previously. 00:49:13.820 --> 00:49:15.140 align:middle line:90% I mean, I'll show this. 00:49:15.140 --> 00:49:16.800 align:middle line:84% So if it's a crystalline material, 00:49:16.800 --> 00:49:20.220 align:middle line:84% you see a set of delta functions indicating 00:49:20.220 --> 00:49:21.680 align:middle line:84% the different coordination shells. 00:49:21.680 --> 00:49:23.055 align:middle line:84% If there's amorphous material, it 00:49:23.055 --> 00:49:26.380 align:middle line:90% gets broadened out a little bit. 00:49:26.380 --> 00:49:30.240 align:middle line:84% Now, I want to give you a more rigorous mathematical, 00:49:30.240 --> 00:49:33.820 align:middle line:84% mathematical definition of a pair distribution function. 00:49:33.820 --> 00:49:34.320 align:middle line:90% Don't worry. 00:49:34.320 --> 00:49:35.560 align:middle line:84% It's not that complicated, actually. 00:49:35.560 --> 00:49:37.740 align:middle line:84% So let's start by looking at the first function 00:49:37.740 --> 00:49:40.460 align:middle line:90% here, which we call rho 1. 00:49:40.460 --> 00:49:42.980 align:middle line:84% Rho 1 is just the probability density 00:49:42.980 --> 00:49:45.080 align:middle line:84% of finding an atom at some particular location. 00:49:45.080 --> 00:49:48.620 align:middle line:90% 00:49:48.620 --> 00:49:51.220 align:middle line:84% Now, in our description here, we are essentially 00:49:51.220 --> 00:49:53.680 align:middle line:84% assuming that atoms are like geometric points. 00:49:53.680 --> 00:49:56.860 align:middle line:84% We can characterize them with a position vector rm. 00:49:56.860 --> 00:50:01.820 align:middle line:84% So the probability of finding atoms at a particular location 00:50:01.820 --> 00:50:04.260 align:middle line:84% would then be a set of delta functions. 00:50:04.260 --> 00:50:06.780 align:middle line:84% If your position vector happened to coincide 00:50:06.780 --> 00:50:10.140 align:middle line:84% with atomic position, then you get one peak. 00:50:10.140 --> 00:50:13.620 align:middle line:84% If you're not at that position of the atoms, then you get 0. 00:50:13.620 --> 00:50:16.740 align:middle line:84% So you expect that this function rho 00:50:16.740 --> 00:50:21.390 align:middle line:84% 1 should just be a sum of a set of delta functions. 00:50:21.390 --> 00:50:23.790 align:middle line:84% And these data functions would only 00:50:23.790 --> 00:50:27.550 align:middle line:84% be non-zero if the argument coincides with one 00:50:27.550 --> 00:50:33.070 align:middle line:90% of the atomic coordinates. 00:50:33.070 --> 00:50:36.290 align:middle line:90% Makes sense? 00:50:36.290 --> 00:50:40.532 align:middle line:90% 00:50:40.532 --> 00:50:42.118 align:middle line:84% It's a probability non-zero if you 00:50:42.118 --> 00:50:43.410 align:middle line:90% happen to hit one of the atoms. 00:50:43.410 --> 00:50:44.630 align:middle line:90% Otherwise, it's 0. 00:50:44.630 --> 00:50:48.230 align:middle line:84% Now, we also put this little bracket here. 00:50:48.230 --> 00:50:50.030 align:middle line:84% The bracket essentially means that we 00:50:50.030 --> 00:50:51.970 align:middle line:90% are doing an ensemble average. 00:50:51.970 --> 00:50:53.810 align:middle line:84% We average across the entire solid material. 00:50:53.810 --> 00:50:57.530 align:middle line:84% So if you do the averaging, then of course, 00:50:57.530 --> 00:51:00.130 align:middle line:84% any kind of this fine structure feature gets smeared out. 00:51:00.130 --> 00:51:05.230 align:middle line:84% What you get is basically just a average density, which is rho. 00:51:05.230 --> 00:51:08.850 align:middle line:84% So rho here is just N divided by V. N is the number of atoms, 00:51:08.850 --> 00:51:11.310 align:middle line:84% V is the volume, so number density. 00:51:11.310 --> 00:51:12.330 align:middle line:90% Makes sense, right? 00:51:12.330 --> 00:51:13.830 align:middle line:84% If it's a homogeneous solid, then it 00:51:13.830 --> 00:51:15.910 align:middle line:90% should be just uniform, simple. 00:51:15.910 --> 00:51:18.390 align:middle line:84% Now, next one, let's look at the probability density 00:51:18.390 --> 00:51:21.510 align:middle line:84% function for finding an atomic pair, one 00:51:21.510 --> 00:51:24.870 align:middle line:84% sitting at this location r, the other sitting at location r 00:51:24.870 --> 00:51:27.650 align:middle line:90% prime. 00:51:27.650 --> 00:51:32.670 align:middle line:84% This is the function we call rho 2. 00:51:32.670 --> 00:51:35.742 align:middle line:84% So similarly, you can also represent this 00:51:35.742 --> 00:51:37.950 align:middle line:84% with a set [? of other ?] product of delta functions. 00:51:37.950 --> 00:51:42.070 align:middle line:84% In order to find the atom-- one at r, the other r prime, 00:51:42.070 --> 00:51:44.670 align:middle line:84% r has to coincide with atomic coordinates for one 00:51:44.670 --> 00:51:45.250 align:middle line:90% of the atoms. 00:51:45.250 --> 00:51:46.750 align:middle line:84% Same thing-- r prime has to coincide 00:51:46.750 --> 00:51:49.310 align:middle line:84% with the atomic coordinates of the other atom. 00:51:49.310 --> 00:51:52.470 align:middle line:84% So that's why mathematically, you 00:51:52.470 --> 00:51:55.650 align:middle line:84% can see that this can be expressed as, again, 00:51:55.650 --> 00:51:58.330 align:middle line:84% the sum of delta functions, and then the product. 00:51:58.330 --> 00:52:01.370 align:middle line:90% 00:52:01.370 --> 00:52:04.310 align:middle line:84% So we have this specific condition here where n cannot be 00:52:04.310 --> 00:52:07.090 align:middle line:84% equal to m because these two atoms cannot coincide. 00:52:07.090 --> 00:52:09.890 align:middle line:84% We are looking for atomic pair, not one single atom OK. 00:52:09.890 --> 00:52:13.590 align:middle line:84% That's why n cannot be equal to m. 00:52:13.590 --> 00:52:14.330 align:middle line:90% Makes sense? 00:52:14.330 --> 00:52:17.390 align:middle line:90% 00:52:17.390 --> 00:52:21.230 align:middle line:84% Now, what we also argue is that if you 00:52:21.230 --> 00:52:24.350 align:middle line:84% have a homogeneous and isotropic solid, 00:52:24.350 --> 00:52:28.350 align:middle line:84% then the magnitude of this function 00:52:28.350 --> 00:52:33.790 align:middle line:84% should not depend on the orientation of the vector of r 00:52:33.790 --> 00:52:34.650 align:middle line:90% minus r prime. 00:52:34.650 --> 00:52:37.410 align:middle line:84% So this is essentially a vector that connects the two points. 00:52:37.410 --> 00:52:40.790 align:middle line:84% If it's isotropic, then the orientation of this vector 00:52:40.790 --> 00:52:42.750 align:middle line:90% really doesn't matter. 00:52:42.750 --> 00:52:46.830 align:middle line:84% Also, if it's homogeneous, then it 00:52:46.830 --> 00:52:51.590 align:middle line:84% doesn't matter where you set the origin point. 00:52:51.590 --> 00:52:55.070 align:middle line:84% And we will argue that this rho 2 should only 00:52:55.070 --> 00:52:59.150 align:middle line:84% depend on the spacing of these two atoms-- 00:52:59.150 --> 00:53:01.470 align:middle line:84% not the specific location of r and r prime, but only 00:53:01.470 --> 00:53:05.230 align:middle line:84% the spacing in between them in a homogeneous isotropic solid. 00:53:05.230 --> 00:53:06.590 align:middle line:90% Makes sense? 00:53:06.590 --> 00:53:10.230 align:middle line:84% So rho 2 should only be a function of the spacing 00:53:10.230 --> 00:53:12.670 align:middle line:90% between the two atoms. 00:53:12.670 --> 00:53:14.670 align:middle line:84% So if you look at now the physical significance, 00:53:14.670 --> 00:53:17.720 align:middle line:84% it basically says that rho 2 is a function that 00:53:17.720 --> 00:53:20.280 align:middle line:84% describes the probability of finding 00:53:20.280 --> 00:53:26.160 align:middle line:84% two atoms with a spacing of r minus r prime, which basically 00:53:26.160 --> 00:53:28.500 align:middle line:84% is the definition of pair distribution function. 00:53:28.500 --> 00:53:32.724 align:middle line:90% 00:53:32.724 --> 00:53:36.480 align:middle line:84% So the only thing we need is to normalize it properly 00:53:36.480 --> 00:53:40.660 align:middle line:84% so that it satisfies these requirement here. 00:53:40.660 --> 00:53:43.540 align:middle line:84% Especially when r goes to infinity, g has to go to unity. 00:53:43.540 --> 00:53:46.080 align:middle line:84% It sets a normalization condition. 00:53:46.080 --> 00:53:49.640 align:middle line:84% So that's why if you look at the rigorous expression here for g, 00:53:49.640 --> 00:53:52.840 align:middle line:84% the partition function is basically rho 2 divided 00:53:52.840 --> 00:53:57.550 align:middle line:84% by rho squared where rho is the average number density. 00:53:57.550 --> 00:53:59.800 align:middle line:84% The reason you do that is because if you look at rho 2 00:53:59.800 --> 00:54:02.600 align:middle line:84% here, it's basically two of the rho 1 functions 00:54:02.600 --> 00:54:05.200 align:middle line:84% multiplied together, practically. 00:54:05.200 --> 00:54:08.880 align:middle line:84% So you imagine that you have to normalize it 00:54:08.880 --> 00:54:11.480 align:middle line:84% by essentially rho 2 squared such that this function, 00:54:11.480 --> 00:54:15.800 align:middle line:84% when r goes to infinity, it should also approach infinity. 00:54:15.800 --> 00:54:20.140 align:middle line:84% So 1 over rho squared is just a normalization coefficient there. 00:54:20.140 --> 00:54:23.960 align:middle line:90% 00:54:23.960 --> 00:54:24.740 align:middle line:90% Makes sense? 00:54:24.740 --> 00:54:28.552 align:middle line:90% 00:54:28.552 --> 00:54:30.760 align:middle line:84% So then what we are trying to do when we are deriving 00:54:30.760 --> 00:54:33.440 align:middle line:84% the diffraction intensity there, the structure factor, 00:54:33.440 --> 00:54:37.200 align:middle line:84% is we are trying to essentially write the structure factor 00:54:37.200 --> 00:54:41.800 align:middle line:84% expression in a way such that it looks similar to this rho 2 00:54:41.800 --> 00:54:42.720 align:middle line:90% function. 00:54:42.720 --> 00:54:45.200 align:middle line:84% That's how we try to build a connection between SQ 00:54:45.200 --> 00:54:47.060 align:middle line:84% and the pair distribution function. 00:54:47.060 --> 00:54:50.680 align:middle line:90% 00:54:50.680 --> 00:54:54.320 align:middle line:84% All right, now here comes the fun part-- 00:54:54.320 --> 00:54:56.505 align:middle line:90% the math. 00:54:56.505 --> 00:54:57.380 align:middle line:90% Don't worry about it. 00:54:57.380 --> 00:54:58.280 align:middle line:90% This looks complicated. 00:54:58.280 --> 00:55:00.400 align:middle line:84% I'm going to go through it step by step to explain 00:55:00.400 --> 00:55:01.760 align:middle line:90% what I'm trying to do here. 00:55:01.760 --> 00:55:06.920 align:middle line:84% All right, so let's come back to this board here. 00:55:06.920 --> 00:55:09.920 align:middle line:84% What I put on the slide is actually not very detailed. 00:55:09.920 --> 00:55:10.980 align:middle line:90% There are a few steps. 00:55:10.980 --> 00:55:12.840 align:middle line:90% Let's skip that. 00:55:12.840 --> 00:55:15.760 align:middle line:84% So let's come back and look at what SQ is. 00:55:15.760 --> 00:55:21.720 align:middle line:90% So we want to solve for SQ 00:55:21.720 --> 00:55:31.400 align:middle line:84% So by definition, SQ is just N fn 00:55:31.400 --> 00:55:38.360 align:middle line:84% squared times the intensity by definition, 00:55:38.360 --> 00:55:42.560 align:middle line:84% times I. This is just the definition of SQ. 00:55:42.560 --> 00:55:45.200 align:middle line:84% So we can expand this I here because we know 00:55:45.200 --> 00:55:47.280 align:middle line:90% what the expression for I is. 00:55:47.280 --> 00:55:49.280 align:middle line:90% So this would just be-- 00:55:49.280 --> 00:55:53.488 align:middle line:84% I'm just going to copy everything here m N 00:55:53.488 --> 00:56:08.474 align:middle line:84% fm e negative iQrn times N n fn e iQrn. 00:56:08.474 --> 00:56:13.850 align:middle line:90% 00:56:13.850 --> 00:56:20.850 align:middle line:84% Now, one more thing I want to do is to add this average. 00:56:20.850 --> 00:56:23.850 align:middle line:90% Why? 00:56:23.850 --> 00:56:26.850 align:middle line:84% Basically, because when we're doing actual experiments. 00:56:26.850 --> 00:56:30.330 align:middle line:84% Practically, we are measuring macroscopic-sized samples. 00:56:30.330 --> 00:56:33.143 align:middle line:84% Also, even though we mentioned that the X-ray is 00:56:33.143 --> 00:56:35.310 align:middle line:84% close to a plane wave, it's not really a plane wave. 00:56:35.310 --> 00:56:38.770 align:middle line:84% There's always a small range of angular dispersion of the X-ray 00:56:38.770 --> 00:56:39.310 align:middle line:90% source. 00:56:39.310 --> 00:56:42.090 align:middle line:84% And the experiment also takes some time. 00:56:42.090 --> 00:56:43.890 align:middle line:84% So it means that during that period, 00:56:43.890 --> 00:56:46.690 align:middle line:84% the atoms are undergoing constant thermal vibration. 00:56:46.690 --> 00:56:48.790 align:middle line:84% So there are lots of factors together, 00:56:48.790 --> 00:56:52.210 align:middle line:84% indicating that we are not measuring one specific snapshot 00:56:52.210 --> 00:56:53.470 align:middle line:90% of atomic positions. 00:56:53.470 --> 00:56:56.610 align:middle line:84% But rather, we are doing some kind of ensemble average 00:56:56.610 --> 00:56:59.630 align:middle line:84% across the solid material for practical experiments. 00:56:59.630 --> 00:57:02.690 align:middle line:84% That's why we put this bracket there. 00:57:02.690 --> 00:57:04.450 align:middle line:90% Makes sense? 00:57:04.450 --> 00:57:05.090 align:middle line:90% All right. 00:57:05.090 --> 00:57:06.610 align:middle line:84% So now if you look at this bracket 00:57:06.610 --> 00:57:10.570 align:middle line:84% here, when I put the bracket, fn basically 00:57:10.570 --> 00:57:13.250 align:middle line:84% is a constant for different atoms. 00:57:13.250 --> 00:57:16.670 align:middle line:84% When you're doing this averaging effectively, 00:57:16.670 --> 00:57:18.570 align:middle line:84% you can pull out the fm essentially 00:57:18.570 --> 00:57:21.450 align:middle line:90% as a factor of average fm. 00:57:21.450 --> 00:57:24.430 align:middle line:84% Same thing-- fn here, when you do the averaging effectively, 00:57:24.430 --> 00:57:27.410 align:middle line:84% you can pull it out as a bracket average fn, which 00:57:27.410 --> 00:57:29.510 align:middle line:90% is the same as average FM 00:57:29.510 --> 00:57:32.810 align:middle line:84% So it means that this factor can actually cancel out 00:57:32.810 --> 00:57:40.030 align:middle line:84% with this one here Once you do the averaging, they cancel out. 00:57:40.030 --> 00:57:42.450 align:middle line:90% So now what do we have? 00:57:42.450 --> 00:57:48.050 align:middle line:84% So now SQ, it makes quite a bit simpler. 00:57:48.050 --> 00:57:51.830 align:middle line:84% You have 1 over N And then you have a-- 00:57:51.830 --> 00:57:56.050 align:middle line:90% 00:57:56.050 --> 00:58:08.410 align:middle line:84% I can take out all the fn there, Qrm, iQrn. 00:58:08.410 --> 00:58:11.690 align:middle line:90% 00:58:11.690 --> 00:58:12.270 align:middle line:90% All right. 00:58:12.270 --> 00:58:15.930 align:middle line:90% 00:58:15.930 --> 00:58:21.370 align:middle line:84% Now, next thing, let's look at these two summations here. 00:58:21.370 --> 00:58:24.130 align:middle line:84% As I said, there are a total of n squared number of terms, 00:58:24.130 --> 00:58:27.410 align:middle line:84% because in parentheses, each contains n terms. 00:58:27.410 --> 00:58:29.530 align:middle line:84% We notice that this n squared terms actually can 00:58:29.530 --> 00:58:31.810 align:middle line:90% be classified into two groups. 00:58:31.810 --> 00:58:35.290 align:middle line:90% One group is when n equals to m. 00:58:35.290 --> 00:58:39.130 align:middle line:84% The other group is when n doesn't equal to m. 00:58:39.130 --> 00:58:42.330 align:middle line:90% What happens if n equals to m? 00:58:42.330 --> 00:58:44.250 align:middle line:84% If n equals to m, you can directly 00:58:44.250 --> 00:58:45.750 align:middle line:84% take the product of these two there, 00:58:45.750 --> 00:58:48.050 align:middle line:84% which is trivial because they cancel 00:58:48.050 --> 00:58:50.090 align:middle line:90% each other, becomes unity. 00:58:50.090 --> 00:58:53.490 align:middle line:84% So each of these terms where n is equal to m essentially 00:58:53.490 --> 00:58:55.370 align:middle line:90% gives you unity. 00:58:55.370 --> 00:58:58.130 align:middle line:84% And you have a total of n capital N 00:58:58.130 --> 00:59:01.250 align:middle line:90% number of those similar terms. 00:59:01.250 --> 00:59:06.800 align:middle line:84% So if total n number of those terms divided by n, 00:59:06.800 --> 00:59:10.780 align:middle line:90% you get unity. 00:59:10.780 --> 00:59:14.980 align:middle line:84% So then what you have, you can further simplify it. 00:59:14.980 --> 00:59:18.220 align:middle line:90% You now have. 00:59:18.220 --> 00:59:26.980 align:middle line:90% m N e negative iQrm times-- 00:59:26.980 --> 00:59:34.856 align:middle line:84% now in summation we can write n does not equal to m, N e iQrn. 00:59:34.856 --> 00:59:44.180 align:middle line:90% 00:59:44.180 --> 00:59:47.700 align:middle line:84% All right, now next part we try to convert it 00:59:47.700 --> 00:59:50.460 align:middle line:84% into a formalism that looks similar to our pattern 00:59:50.460 --> 00:59:52.820 align:middle line:84% distribution function, rho 2 function. 00:59:52.820 --> 00:59:54.000 align:middle line:90% How do we do that? 00:59:54.000 --> 00:59:58.500 align:middle line:84% We have to use this unique property of delta functions 00:59:58.500 --> 01:00:01.960 align:middle line:84% So if you think about any arbitrary function-- 01:00:01.960 --> 01:00:04.500 align:middle line:84% any kind of, let's say, any arbitrary function. 01:00:04.500 --> 01:00:06.820 align:middle line:84% If you integrate from positive infinity 01:00:06.820 --> 01:00:10.840 align:middle line:84% to negative infinity for any kind of arbitrary function, 01:00:10.840 --> 01:00:21.380 align:middle line:84% let's say Fx times a delta function of x minus a dx. 01:00:21.380 --> 01:00:24.060 align:middle line:90% 01:00:24.060 --> 01:00:24.880 align:middle line:90% What do you get? 01:00:24.880 --> 01:00:29.340 align:middle line:90% 01:00:29.340 --> 01:00:31.420 align:middle line:90% You're doing integration. 01:00:31.420 --> 01:00:33.040 align:middle line:90% It contains delta function. 01:00:33.040 --> 01:00:37.680 align:middle line:84% Delta function is 0 everywhere except when x minus a equals 0. 01:00:37.680 --> 01:00:41.700 align:middle line:90% 01:00:41.700 --> 01:00:48.580 align:middle line:84% So that means when x minus a equals to 0, then x equals to a. 01:00:48.580 --> 01:00:52.220 align:middle line:84% At that point, the integration becomes Fa, and everywhere else 01:00:52.220 --> 01:00:53.540 align:middle line:90% is basically zero. 01:00:53.540 --> 01:00:57.980 align:middle line:90% So this should be equal to a F-- 01:00:57.980 --> 01:00:59.240 align:middle line:90% I don't know. 01:00:59.240 --> 01:01:00.780 align:middle line:84% I'll write that on this side here. 01:01:00.780 --> 01:01:04.620 align:middle line:90% 01:01:04.620 --> 01:01:10.960 align:middle line:84% Fa-- this is a property of delta function. 01:01:10.960 --> 01:01:19.688 align:middle line:90% 01:01:19.688 --> 01:01:21.137 align:middle line:90% [LAUGHTER] 01:01:21.137 --> 01:01:23.220 align:middle line:84% This is actually the definition of delta function. 01:01:23.220 --> 01:01:25.060 align:middle line:84% Delta functions, if you integrate from 0 to infinity, 01:01:25.060 --> 01:01:26.220 align:middle line:90% it's equal to unity. 01:01:26.220 --> 01:01:31.580 align:middle line:84% And the function is the only non-zero at the 0 there. 01:01:31.580 --> 01:01:33.500 align:middle line:84% All right, so now next thing we want to do is, 01:01:33.500 --> 01:01:40.140 align:middle line:84% let's take this function Fa as e. 01:01:40.140 --> 01:01:48.540 align:middle line:84% Essentially, our function f is basically e, 01:01:48.540 --> 01:01:49.860 align:middle line:90% for example, negative iQrm. 01:01:49.860 --> 01:01:54.260 align:middle line:90% 01:01:54.260 --> 01:01:55.340 align:middle line:90% We take this function. 01:01:55.340 --> 01:01:58.950 align:middle line:90% 01:01:58.950 --> 01:02:05.430 align:middle line:84% And then we take a essentially as our rm, 01:02:05.430 --> 01:02:07.650 align:middle line:90% and x essentially as r. 01:02:07.650 --> 01:02:10.733 align:middle line:90% 01:02:10.733 --> 01:02:11.650 align:middle line:90% Let's see what we get. 01:02:11.650 --> 01:02:15.870 align:middle line:90% So now Fa will becomes e-- 01:02:15.870 --> 01:02:22.190 align:middle line:84% basically negative iQrm This should 01:02:22.190 --> 01:02:28.070 align:middle line:84% equal to integral of the function itself, which is 01:02:28.070 --> 01:03:01.230 align:middle line:84% e negative iQr times the delta function r minus rm dr. 01:03:01.230 --> 01:03:04.470 align:middle line:84% We can actually also do the same thing for rn. 01:03:04.470 --> 01:03:06.290 align:middle line:90% I will also write it here-- 01:03:06.290 --> 01:03:06.790 align:middle line:90% iQrn. 01:03:06.790 --> 01:03:09.630 align:middle line:90% 01:03:09.630 --> 01:03:16.110 align:middle line:90% This is a similar e iQr. 01:03:16.110 --> 01:03:20.230 align:middle line:84% Now to be different here, I use r prime, 01:03:20.230 --> 01:03:23.430 align:middle line:84% Just to differentiate between the two different integration 01:03:23.430 --> 01:03:24.670 align:middle line:90% variables here. 01:03:24.670 --> 01:03:31.890 align:middle line:84% This is delta r prime minus rn dr prime. 01:03:31.890 --> 01:03:41.510 align:middle line:90% 01:03:41.510 --> 01:03:43.750 align:middle line:84% Essentially, you're integrating this function 01:03:43.750 --> 01:03:46.270 align:middle line:90% over the whole space. 01:03:46.270 --> 01:03:49.430 align:middle line:84% And the delta function is only non-zero when r equals rm. 01:03:49.430 --> 01:03:52.870 align:middle line:84% So what you do is then you plug rm into the equation, 01:03:52.870 --> 01:03:55.080 align:middle line:90% the function there. 01:03:55.080 --> 01:03:56.500 align:middle line:90% Yeah, same thing with this one. 01:03:56.500 --> 01:03:59.220 align:middle line:84% I just changed the integration argument, 01:03:59.220 --> 01:04:02.840 align:middle line:84% so it's less confusing as you'll see. 01:04:02.840 --> 01:04:10.420 align:middle line:84% So then what we do, let's plug this back to our equation here. 01:04:10.420 --> 01:04:16.040 align:middle line:90% 01:04:16.040 --> 01:04:17.700 align:middle line:90% I'm still running out of space. 01:04:17.700 --> 01:04:22.000 align:middle line:90% I think 1 over N. 01:04:22.000 --> 01:04:23.560 align:middle line:90% So then what do you have here? 01:04:23.560 --> 01:04:24.080 align:middle line:90% This one. 01:04:24.080 --> 01:04:28.640 align:middle line:90% 01:04:28.640 --> 01:04:37.480 align:middle line:84% You have integration of e negative iQr 01:04:37.480 --> 01:04:46.120 align:middle line:84% times delta function r minus rm dr, 01:04:46.120 --> 01:04:52.260 align:middle line:90% then times another integration-- 01:04:52.260 --> 01:04:55.040 align:middle line:90% 01:04:55.040 --> 01:04:55.880 align:middle line:90% e. 01:04:55.880 --> 01:04:57.052 align:middle line:90% Sorry, I need to do some-- 01:04:57.052 --> 01:04:58.260 align:middle line:90% I forgot the summation there. 01:04:58.260 --> 01:05:02.680 align:middle line:90% I need to sum over m. 01:05:02.680 --> 01:05:05.140 align:middle line:90% Here, I need to sum over n. 01:05:05.140 --> 01:05:14.240 align:middle line:90% 01:05:14.240 --> 01:05:23.000 align:middle line:84% And I also have another integration e iQr prime times 01:05:23.000 --> 01:05:29.500 align:middle line:84% delta function r prime minus rn dr prime. 01:05:29.500 --> 01:05:37.788 align:middle line:90% 01:05:37.788 --> 01:05:40.680 align:middle line:90% It looks messy. 01:05:40.680 --> 01:05:44.120 align:middle line:84% It's fine, because what we can do, let's see what we have here. 01:05:44.120 --> 01:05:46.560 align:middle line:90% We essentially have an integral. 01:05:46.560 --> 01:05:50.960 align:middle line:84% There are a total of N terms in the integral there. 01:05:50.960 --> 01:05:55.880 align:middle line:84% It's a sum of integral times a sum of integral. 01:05:55.880 --> 01:05:57.020 align:middle line:90% This is a linear operation. 01:05:57.020 --> 01:05:58.600 align:middle line:90% Summation is a linear operation. 01:05:58.600 --> 01:06:01.900 align:middle line:84% So what you can do, because it's linear, 01:06:01.900 --> 01:06:06.040 align:middle line:84% you can essentially move this summation into the integration 01:06:06.040 --> 01:06:06.540 align:middle line:90% there. 01:06:06.540 --> 01:06:07.123 align:middle line:90% It's the same. 01:06:07.123 --> 01:06:10.640 align:middle line:90% 01:06:10.640 --> 01:06:15.520 align:middle line:84% So then what you do, you can then do a double integral first, 01:06:15.520 --> 01:06:18.823 align:middle line:90% and then you do the summation. 01:06:18.823 --> 01:06:20.740 align:middle line:84% Yeah, I don't know if I still need to copy it. 01:06:20.740 --> 01:06:24.940 align:middle line:84% But I mean, basically, it then becomes this guy. 01:06:24.940 --> 01:06:29.320 align:middle line:90% 01:06:29.320 --> 01:06:40.240 align:middle line:84% And also because this term here e iQr and e iQr prime, 01:06:40.240 --> 01:06:42.720 align:middle line:84% you can combine them together to get this one 01:06:42.720 --> 01:06:45.660 align:middle line:90% here, e iQr minus r prime. 01:06:45.660 --> 01:06:47.060 align:middle line:90% This was combined here. 01:06:47.060 --> 01:06:50.090 align:middle line:84% And then you essentially move the summation sign 01:06:50.090 --> 01:06:51.070 align:middle line:90% into the integral. 01:06:51.070 --> 01:06:53.890 align:middle line:90% 01:06:53.890 --> 01:06:56.290 align:middle line:84% Everything after this is actually pretty straightforward 01:06:56.290 --> 01:07:03.350 align:middle line:84% because this part here is exactly our rho 2. 01:07:03.350 --> 01:07:08.690 align:middle line:90% 01:07:08.690 --> 01:07:10.990 align:middle line:84% All this is basically just rho 2. 01:07:10.990 --> 01:07:15.810 align:middle line:90% 01:07:15.810 --> 01:07:18.670 align:middle line:84% So I guess I don't have to write it down. 01:07:18.670 --> 01:07:20.870 align:middle line:84% I just explain what we have been doing here. 01:07:20.870 --> 01:07:24.930 align:middle line:90% 01:07:24.930 --> 01:07:27.710 align:middle line:84% OK, so then the next couple of steps, 01:07:27.710 --> 01:07:30.010 align:middle line:84% essentially, you just try to clean up. 01:07:30.010 --> 01:07:33.890 align:middle line:84% You essentially put in a proper normalization constant. 01:07:33.890 --> 01:07:37.650 align:middle line:84% So you write rho 2 now as g, which is our pair distribution 01:07:37.650 --> 01:07:38.650 align:middle line:90% function. 01:07:38.650 --> 01:07:41.072 align:middle line:84% And you get this kind of expression here 01:07:41.072 --> 01:07:42.530 align:middle line:84% Now, if you look at the expression, 01:07:42.530 --> 01:07:45.770 align:middle line:84% you have a complex exponential times g. 01:07:45.770 --> 01:07:47.350 align:middle line:90% Then you do the integration. 01:07:47.350 --> 01:07:52.690 align:middle line:84% This is basically the Fourier transform of gr. 01:07:52.690 --> 01:07:55.490 align:middle line:84% So basically what it says is that the structure factor 01:07:55.490 --> 01:07:58.410 align:middle line:84% directly correlates with the Fourier transform of the pair 01:07:58.410 --> 01:08:01.490 align:middle line:90% distribution function. 01:08:01.490 --> 01:08:03.430 align:middle line:84% That's the takeaway message on the slide here. 01:08:03.430 --> 01:08:06.572 align:middle line:90% 01:08:06.572 --> 01:08:08.530 align:middle line:84% That's a key takeaway message, structure factor 01:08:08.530 --> 01:08:11.890 align:middle line:84% relates to Fourier transform of the pair distribution function. 01:08:11.890 --> 01:08:13.707 align:middle line:84% And you can mathematically further simplify 01:08:13.707 --> 01:08:15.290 align:middle line:84% the equation a little bit, because you 01:08:15.290 --> 01:08:18.850 align:middle line:84% notice in this integration here, you still 01:08:18.850 --> 01:08:24.210 align:middle line:84% have to integrate over this vector here. 01:08:24.210 --> 01:08:25.587 align:middle line:90% R is a vector. 01:08:25.587 --> 01:08:27.170 align:middle line:84% You have to integrate over essentially 01:08:27.170 --> 01:08:28.710 align:middle line:90% a three-dimensional space r. 01:08:28.710 --> 01:08:30.930 align:middle line:84% But if it's an isotropic solid, you 01:08:30.930 --> 01:08:34.267 align:middle line:84% can imagine that it's only the magnitude of r that matters. 01:08:34.267 --> 01:08:35.850 align:middle line:84% Its orientation doesn't really matter. 01:08:35.850 --> 01:08:38.370 align:middle line:84% So you can then do the integration essentially 01:08:38.370 --> 01:08:42.010 align:middle line:84% in a spherical coordinate which allows you to further simplify 01:08:42.010 --> 01:08:42.670 align:middle line:90% the equation. 01:08:42.670 --> 01:08:44.212 align:middle line:84% I'm not going to go through the math, 01:08:44.212 --> 01:08:47.450 align:middle line:84% but this is what eventually comes out in isotropic material. 01:08:47.450 --> 01:08:49.930 align:middle line:84% Then SQ will then directly correlate 01:08:49.930 --> 01:08:52.210 align:middle line:84% with this one-dimensional integration over now 01:08:52.210 --> 01:08:56.090 align:middle line:84% a scalar r, rather than a vector r. 01:08:56.090 --> 01:08:57.750 align:middle line:84% You can then do the inverse transform, 01:08:57.750 --> 01:09:00.770 align:middle line:84% which tells you this is how you relate the pair distribution 01:09:00.770 --> 01:09:06.410 align:middle line:84% function to the structure vector in an isotropic, homogeneous 01:09:06.410 --> 01:09:07.470 align:middle line:90% amorphous material. 01:09:07.470 --> 01:09:16.250 align:middle line:90% 01:09:16.250 --> 01:09:20.529 align:middle line:84% So bottom line, if you can measure the diffraction spectra 01:09:20.529 --> 01:09:23.689 align:middle line:84% of a material, which usually is given in terms of intensity I 01:09:23.689 --> 01:09:31.529 align:middle line:84% as a function of theta, you then convert the I theta into SQ. 01:09:31.529 --> 01:09:36.090 align:middle line:84% You can then directly calculate what is the corresponding pair 01:09:36.090 --> 01:09:37.149 align:middle line:90% distribution function. 01:09:37.149 --> 01:09:39.689 align:middle line:90% 01:09:39.689 --> 01:09:43.420 align:middle line:84% By the way, this is known as the Debye scattering equation. 01:09:43.420 --> 01:09:47.220 align:middle line:84% It shows that for this kind of even seemingly featureless 01:09:47.220 --> 01:09:49.180 align:middle line:84% broad diffraction spectra, you can still 01:09:49.180 --> 01:09:52.003 align:middle line:84% extract very useful quantitative information, in this case, pair 01:09:52.003 --> 01:09:54.420 align:middle line:84% distribution function, to describe the amorphous structure 01:09:54.420 --> 01:09:54.920 align:middle line:90% there. 01:09:54.920 --> 01:09:58.800 align:middle line:90% 01:09:58.800 --> 01:10:01.120 align:middle line:84% Now, that may seem simple enough. 01:10:01.120 --> 01:10:02.880 align:middle line:84% You can just get your glass sample, 01:10:02.880 --> 01:10:05.880 align:middle line:84% go and do X-ray diffraction, and then come back and get a PDF. 01:10:05.880 --> 01:10:09.000 align:middle line:84% It's actually quite a bit more tricky than that. 01:10:09.000 --> 01:10:10.120 align:middle line:90% We don't have enough time. 01:10:10.120 --> 01:10:13.340 align:middle line:84% In the next lecture, we're going to talk about how you actually 01:10:13.340 --> 01:10:15.900 align:middle line:84% correct your experimental data to really implement 01:10:15.900 --> 01:10:16.680 align:middle line:90% this equation. 01:10:16.680 --> 01:10:19.740 align:middle line:90% It's non-trivial. 01:10:19.740 --> 01:10:22.380 align:middle line:84% So I will stop here for this lecture. 01:10:22.380 --> 01:10:25.050 align:middle line:90% All right, thank you. 01:10:25.050 --> 01:10:42.000 align:middle line:90%